

## Tolliver's Travels

## ...with Mr. Mouse in the Worldwide Web

A Book of Entertainments and Diversions

by

### Fred Stang

Smashwords Edition

Copyright © 2016 by Fred Stang

All Rights Reserved.

### Table of Contents

Preface

1. Into the World-Wide Web

2. The Identical Men

3. The Evil King and the Fair Maiden

4. A Stern Teacher and His Brilliant Young Pupil

5. In the Shadow of the Great Pyramid

6. The Grumpy Camel and Thousands of Bananas

7. Two Timers – The Hard-Boiled Egg Story

8. The Case of the Missing Dollar

9. Apples and Oranges

10. The Big Cheese

11. The Friendly Isle of Amiti

12. The Truth of the Matter

13. Between the Lions

14. Which Doctor?

The Island of Doctor Yes, Doctor No, and Doctor Maybe

15. Mr Quirm

Many a Quaint and Curious Volume of Forgotten Lore

16. Chessboard and Dominoes

17. No Big Deal

18. Pick a Number, Any Number

19. The Monte Hall Paradox

20. Reflections on the Cigar Game

21. Ping-Pong Ball... and Pins Galore

22. Same Place, Same Time

23. Farewell, Mouse

### PREFACE

Here, presented for your entertainment, are the adventures of Tolliver Tilden III in the World-Wide Web, a tangle of puzzling encounters, all accompanied by his new friend and helper, Mr Mouse. Unlike many storybooks, you can read only the parts you are enjoying and be welcome to skip what does not catch your interest. But, a brief word about the reading of it.

These challenges are presented one by one as the pair travel along in their virtual world, all in all twenty questions, more or less. At a certain point, in each and every case, you will see that the challenge has been presented and it is clear what is to be discovered, found out, or devised as a solution. At this point, you are free to put down the work and find this solution on your own. Or, you can simply keep reading, to find clues and hints perhaps, and to read the solution proposed. You will not have to "look up" an answer!

It is suggested that more fun can be had by taking the time to find the answer for yourself, but this is entirely up to you. The first commandment for you here is "enjoy!"

### CHAPTER 1

INTO THE WORLD WIDE WEB

On his twelfth birthday, Tolliver Tilden III turned on his present, a brand-new shining computer, and was promptly swallowed by the World-Wide Web. It was the first time he had ever worked a computer or gone online all by himself, a very exciting moment, but what an unexpected outcome!

A deep whooshing sound transported him through a swirling maze of zips, zings and zwiggles, flashing lights and gossamer wires, depositing him in the middle of a country road facing toward a low hill. And that was odd because, only moments earlier, he had been thinking about just such a road, with a view of just such a hill.

He heard someone singing:

"Happy, happy are the mice.

happy are the nice, nice,

nice, nice mice..."

Before him, something he had most definitely _not_ been thinking of, cavorted a mouse, about four feet tall, roundish in figure, and wearing the most unmatched outfit Tollie had ever seen – a plaid jacket, red and white striped shirt, polka dot tie (yellow on black), and checkered pants in purple and orange. The toes of his natty black shoes curled upward. The shoes were themselves impossibly well-polished and glinted giddily over his green socks.

"Hello, I am Mouse, but you may call me Mr Mouse." He spoke! He also stood upright on his hind legs and walked on these as easily as a human walks on two feet. "I am here to serve you, Tollie."

"How do you know my name?" Tollie was a very curious boy, having caused his parents no end of dismay in having to answer his endless questions day in and day out, though they miraculously remained patient with him. To themselves, they silently (and optimistically) hoped he was just going through a "phase."

"Ah! I am your personal mouse, here to take you wherever you want to go. Of course your name was given to me before you even came here."

"I was wondering," Tollie said, "how I landed exactly here, when this was what was in my mind, just before I got here?"

"You will find," Mr Mouse told him in a gentle, patient manner, "that this is how things work here in the World-Wide Web. You will end up in the middle of many things that you only just thought of, and also many that your natural curiosity will just lead you to. You'll get used to it."

Quite a contrast to Tollie's first impression of the cavorting, singing mouse. "Are you in charge here?" he asked.

Mr Mouse considered this as if the thought of someone being "in charge" of the World-Wide Web had never entered his mind before.

"Noooo... I couldn't say that... Maybe we will find out together just who is in charge, if anyone is." He looked genuinely perplexed.

### CHAPTER 2

THE IDENTICAL MEN

As if on cue, two roly-poly gentlemen came jaunting over the hill toward them. You could tell they were gentlemen as they wore identical vests and glasses. But Tollie _knew_ for certain that they were not in charge! They were altogether too silly.

And they were identical, both with wide brown eyes, straight black hair that ended in bangs just above their eyes, and smiles that showed their gleaming white teeth to good advantage.

"They're twins," Tollie whispered to Mr Mouse.

"NOT!" Mouse whispered right back. "I have encountered them before."

The first man said in a jolly way, "Hi, I'm Hop Hop," and, to illustrate, did a little hop on his left foot.

"And I'm Hip Hip," said the other, performing a couple of little bumps with his right hip to illustrate.

Tollie stifled a laugh. "And I'm Tollie," he introduced himself, tossing his head back and forth so that he, too, had his own signature gesture.

"Hi, Tollie," they both answered breezily. "And hello, Mr Mouse, we meet again."

"Mr Mouse tells me you are not twins." Tollie could not keep his curiosity in. "But you look so alike! How can that be?"

"Well, we are identical and born of the same woman at the same time of the same day of the same year," Hop Hop answered.

" _And,_ we are not twins!" Hip Hip chimed in.

They both giggled merrily, seeing the puzzlement in Tollie's face.

"How can it be?" Tollie wondered out loud.

"Think, Tollie," Mr Mouse said gently and patiently, as Tollie came to learn was his way. "There is a possible explanation. Think what they could be, if not twins."

"But they have to be twins!" Tollie protested. "If everything they said is true, about being born the same time, then they must be twins."

"Tollie," Mr Mouse went on, "if _everything_ they said is true, then it is also true that they are not twins, as they said."

Tollie could not argue with this logic and thought and thought, long and hard, and then some, his face scrunched in concentration, as was his way. How can it be? How can it be?

Indeed, how can it be?

After a good amount of thinking, he was ready to give up... but...

Over the hill now came a third roly-poly gentleman, and he, too, was identical to the first two!

He did a little hop and then a little hip bump. "Hi, I'm Hip Hop," he announced. The three of them stood together sharing their mirth at Tollie's expression.

"Well, of course you are!" Tollie said, laughing with them.

Understanding came visibly to him as it dawned on him what Mr Mouse had hinted at.

"You're triplets!"

"As you say, so we are," Hop Hop, Hip Hip and Hip Hop said in unison. They hopped, hipped and hip-hopped away, waving goodbye in unison as they went their merry way.

"The first two men were not twins, but two of triplets," Mr Mouse summarized, as Tollie would come to learn was also his way. "There is an explanation, as I promised. I will not lead you astray, Tollie. All your challenges will be fair. But you will have to think!"

Tollie shook Mr Mouse's gray "hand" and nodded, smiling. "Very good, and I'll agree to that. Where to next?"

"That is up to you, Tollie. I go where your thoughts take us. Just put your hand on my back and we will go!"

Tollie did so.

### CHAPTER 3

THE EVIL KING AND THE FAIR MAIDEN

This is how they ended up in the old country, for that was what had entered Tollie's mind just then.

Here in a beautiful village from the days of yore a sad tale unfolded. There lived a lovely maiden with her parents in a simple but cozy cottage. They were not rich, but had enough, and were happy people in a happy family of three, healthy and hard-working farmers.

All was well had not the king, riding back from the day's hunt, spied the comely lass. He was smitten by her natural charms and decided then and there that she would become his wife. But the maiden did not return his interest – she did not want to marry the king. It would bring her great riches, but she did not and could not ever love him.

So he decreed it. The king had it announced that she would have a choice. This would make it seem to his subjects that he was actually being fair.

She would be presented with a bag of 100 marbles, 99 of which would be white and one red. Out of this bag, without looking into its contents, the maiden was to select one marble. If she selected the red marble she would not have to marry the king. But if she picked a white marble, she would have to agree to wed him or the king would take the cottage away from her parents, leaving them bereft of a home and the work they did in the fields that sustained them. The maiden was to make her selection from the bag on the morrow.

That night the king consulted the Three Wicked Witches of the Other Direction, his trusted advisors. The horrible hags, jealous anyway of the maiden's beauty, churned their cauldron through the night, in the darkness of their cave. "Double, double, toil and trouble, fire burn and cauldron bubble," they chanted, their evil minds working to find a way the king could guarantee that the maiden would not even have her one-in-a-hundred chance to pull out the one red marble.

In the morning, they told the king their plan. He was to put only white marbles in the bag. After all, he was the king. No one would dare check. The maiden could only pull out a white marble.

"How unfair this all is," Tollie lamented.

"Well," said Mr Mouse gallantly, "the maiden might find a way. But what could it be? Can you think of a way for her to save herself?"

Tollie scrunched up his face in deep thought as the scene continued to unfold before them. How could the maiden escape the designs of the king and the devious plan devised by the Three Wicked Witches of the Other Direction?

Now they watched as, during the night, the maiden had a dream. It being the World-Wide Web, Tollie and Mr Mouse could see her dream as the maiden slept. She dreamt that her would-be, one day, Prince Charming, whom she had not yet met but whom she recognized at once, visited her and whispered in her ear, telling her of the plan of the Witches and what she was to do the next day to get out of having to marry the king and to save her and her parents' home. But, though the Web let them see her dreaming, it did not allow them to hear what he whispered, try as they might.

The appointed time came on the morrow and, at noon, the entire village showed up on the white gravel of the central square, to watch the maiden make her choice.

There was much fanfare as the king arrived, then the maiden. A hushed silence fell as the king handed her the bag of marbles, which she took from him calmly enough. Tollie watched, wide-eyed and feeling helpless, wondering what she could do. She did not have a chance!

Or did she? Mr Mouse had said all the challenges would be fair, so it seemed there must be some way she could thwart the king.

Smiling at the king and then at all the villagers, the maiden reached into the bag and pulled out a marble, holding it within her hand so no one could see its color!

Tollie tensed up, but Mr Mouse seemed at ease.

The maiden forthwith dropped the selected marble into her purse, still concealing it so no one, villager or king, could see its color even now. She then emptied the king's bag of its remaining 99 marbles, spilling them out onto the white gravel of the village square.

All the marbles were clearly white, not a red one in sight.

"I must have picked the red marble," she announced so all could hear, "for there was no red marble remaining in the bag." She waved the empty bag to demonstrate.

The king was furious, but there was nothing for him to do without revealing to all that there had never been a red marble in the bag, giving away his devious plot.

The maiden smiled and left with her parents, all quite relieved. And, in time, the maiden and her family moved to a wondrous land far away (where there were no kings), never to worry about the evil king again, and, of course, she met her Prince Charming and they all lived happily ever after.

"Whew!" Tollie let out his breath. "She did figure out a way. Amazing!"

"Either she was clever or her future Prince Charming rescued her by giving her the plan," said Mr Mouse. "Let us hope the king has learned something today."

That worthy was stalking away in haste, embarrassed to face the crowd any longer than he had to. But they all knew what had happened. Behind his back they would always celebrate, among themselves only, the failure of his plan to force the maiden into marrying him.

"Good thing," Tollie said. "They are a nice family. They are nobler than their king has been. I hope they will be happy in their future home and that the king will in time become a more benevolent ruler, unlikely as that might seem."

Tollie put his hand on Mouse's back without really thinking about it. (He had taken to thinking of him as "Mouse," though he wouldn't dream of saying it out loud.) It seemed natural to him now that Mouse could transport him to whatever time and place his flights of fancy and whim led them. He was thinking how glad he was that, so far, their adventures had not involved too much math. He was a bit intimidated by the subject. Little did he know!

### CHAPTER 4

A STERN TEACHER AND HIS BRILLIANT YOUNG PUPIL

He had no sooner thought it than they ended up in a mathematics classroom of centuries ago, facing a stern-looking teacher, a man of evident sour disposition.

Tollie vowed that in future he would try harder to keep his thoughts under control. Mouse was chuckling merrily. "You don't always end up exactly where you thought you would."

"Not quite," Tollie acknowledged, somewhat grumpily. But he had to admit to himself it was not Mouse's fault. He had been thinking the thoughts that led them here, even though this was not what he had in mind.

"Mr Mouse, can we leave here now? It was a mistake, my thinking of math and math classrooms."

"Oh, no, Tollie, we have to face our challenges," Mouse answered cheerily (altogether too cheerily for Tollie's mood). "Curiosity killed the cat, you know."

Mouse said this last with what seemed to Tollie to be particular relish. Cats, Tollie reasoned, were doubtless not Mouse's favorite creatures.

They hovered at the edge of the classroom, not exactly participants, but observers nevertheless. One of the young pupils, now just a boy, but a boy with a big destiny, was Carl Friedrich Gauss. Tollie had read of him, mentioned along with Archimedes and Sir Isaac Newton as one of the greatest mathematicians of all time. Tollie was awed by this turn of events, even though he knew he was mostly responsible for it.

"Have you figured out who is in charge of the World-Wide Web, Tollie?" Mouse asked mischievously. It made Tollie think he had an idea about that, but they were distracted by the teacher droning monotonously at the students.

"Now," the sour teacher said, "you will all add up the numbers going from one to one-hundred, write down the answer on your tablets, and bring them up to me. Just put your answers down on my desk." Of course, he simply did not feel like teaching that day and this "exercise" was his excuse to sit back and relax while the pupils toiled over the meaningless task he had set.

Boring, Tollie thought and wished he had never thought of mathematics at all. This was going to take forever.

"Not forever," Mouse said with a smile, seeming to read his mind. "Mathematics can be a lot of fun, if you let it."

"Yeah, yeah," Tollie grumbled.

But then he watched curiously and not without interest as the young Carl Friedrich walked only moments later to the front of the classroom and placed his tablet on the teacher's desk. It had taken him only seconds. "Tollie," Mouse whispered, "how do you think he added up all those numbers, from one to one-hundred, so quickly?" Mouse, of course, had no doubt that Carl Friedrich had come up with the right answer, even though it seemed to have come to him miraculously quickly.

It was all giving Tollie a headache. He could not figure it out. Of course, not believing it was possible, he could not find his way to an answer. (There was much he had still to learn from Mouse.)

The teacher frowned at young Gauss, also not believing he had the right answer so quickly. He did not deign to look at the tablet but waited and waited, until all the pupils brought their tablets to his desk in dribs and drabbles, laying them on top of Carl Friedrich Gauss's tablet, so his was on the bottom, in what seemed to Tollie like an eternity.

Teacher went through the tablets one by one, then called out the names of the pupils who had the right answer, only about five out of thirty or so students in the class. He turned over the final tablet, Carl's, his expression changing to one of amazement.

The answer was right. But how had this young boy done it so quickly? He begrudgingly called out Carl Friedrich Gauss's name as having the correct answer.

"So how did he get the answer just like that?" Tollie asked Mouse.

He scrunched up his face, deep in thought.

Mouse tried to help him out. "Think of a _mathematical_ way to get the answer instead of just adding up the numbers one-by-one. Think of the list of numbers itself. Can you see anything about it that could lead to such a quick answer? There's more than one way to skin a cat!" Again, Mouse seemed to relish the last – he and cats would never be an item, it was clear.

Mouse was generous with his hints, but Tollie could not get over his awe of young Carl Friedrich's amazing feat. He just did not see how it was possible.

To his astonishment, Carl himself walked over to them, seeming to see them while no one else had (for he truly had uncanny ability), and introduced himself. Mouse and Tollie greeted him warmly. Of course, the curious Tollie could hardly wait. "How did you do it, Carl?"

"It was a matter of pairing up the numbers," Carl explained. "You have one-hundred numbers in sequence from one to one-hundred and you have to find the sum of all of these. I saw at once that you could pair up the numbers into fifty pairs in a certain way."

Tollie was beginning to get an idea, but it still did not fully dawn upon him. Mouse, of course, already knew the answer, which Tollie found irritating at times.

"You can make a pair out of the highest and the lowest numbers," Carl went on. "That is, 1 and 100, which add up to 101. The same is true of the second highest and lowest numbers, so that 2 and 99 also add up to 101, and the next highest and lowest, 3 and 98, also add up to 101, right down to 50 and 51, which add up also to 101. This is a process called extrapolation, the extending out of a pattern that proceeds in a logical way. If one step is true, the next step will also be true, and the one after that, and so on.

"Soooo, what you have are 50 pairs of numbers that add up to 101. The total would be 50 times 101, which gives you 5050."

Tollie now understood and was chagrined that he had not thought of this himself. "You make it seem so easy, Carl."

Carl smiled at Tollie. "I always believe there are answers, it is just a matter of finding them. But believing in the reality of there being an answer is always the first step. Even if there is not an answer, you have to show why not. And you can think your way through it, by patiently applying logic and reason to the question at hand."

"I see, somewhat," Tollie said. "I didn't really believe there was an answer to this one so that worked against me finding one. And, _then,_ I did not think to _prove_ that an answer was not possible, which would itself be an answer of sorts."

"Exactly," Carl said, his eyes twinkling. He took his leave of them. "I'm supposed to be sitting at my desk, but it was fun meeting you, Tollie. I think your natural curiosity will make you a better mathematician than you think right now."

"And it was fun meeting you, too. You taught me to believe in and really seek answers even when it seems at first to be impossible. Thank you!"

"You're very welcome." Carl gave a slight bow, which both Tollie and Mouse returned, and wandered back to his desk.

"That was wonderful, Mr Mouse. There is so much to learn here!"

"Oh, yes, Tollie, it is a big world. Not quite like your everyday world, is it?"

"But somehow similar."

"Oh, yes, that it is," Mouse said wistfully, for it was the only world he existed in, after all. "Just remember, not everything you find here is the exact truth. For example, this incident with Carl and his teacher probably never really happened, but when enough people believe it happened, it can be real here in the World-Wide Web. We don't really know, but it's doubtful and many think it is a made-up story. But it is a good story and that is why it is real here. Just take things with a grain of salt.

"Nevertheless," Mouse added, "what Carl told you is quite true and real. You do need to believe that there are answers and that you can find them. And when there is not an answer you can show why not. And, remember, when I give you a challenge it will always be an honest one, and there will be an answer that you will be able to figure out."

"But I haven't figured out any of them so far," Tollie protested.

"You are still learning, and you are getting better at it. Did you notice how quickly you caught on when Carl explained his method to you?"

"Sort of," Tollie said, still doubtful.

"You will see," Mouse continued confidently. "Shall we go? Just let me know where..."

Tollie put his hand on Mouse's back and thought he would like to see the Great Pyramid in Egypt!

### CHAPTER 5

IN THE SHADOW OF THE GREAT PYRAMID

And they were there.

"This takes some getting used to, Mr Mouse!"

They were at the foot of the Great Pyramid and beholding it with a certain healthy respect. Seeing the real thing gave Tollie the true measure of it. It was, simply, 'huge.'

It inspired Mouse to sing:

"Happy, happy are the mice.

Happy are the nice, nice,

nice, nice mice..."

He danced around joyously. There was in front of them also, of course, the Sphinx, a cat of sorts, practically in the shadow of the pyramid, and Mouse seemed to flaunt his "mouseness" (or "mousity"), aware that the Sphinx was only a statue and could do him no harm. Tollie chuckled to himself.

They heard another voice singing, joining in with the same tune as Mouse's signature ditty.

"I am, amid a pyramid,

amid a pyramid, mid

pyramid..."

Materializing before them was an Ancient Egyptian. This seemed sudden to them, but it was only because this was the moment the Ancient Egyptian had chosen to reveal his profile to them! You see, he was completely flat and invisible when he faced you. Only when he turned his side to you in profile could you see him.

Tollie had thought it was just that the ancient Egyptians had not mastered perspective and could not represent three dimensions on a flat surface in their art, so they appeared two-dimensional. But here, they actually were flat and only visible in profile.

"They really were flat?" Tollie asked incredulously, and Mouse grinned broadly.

"That is the only way we can see them here."

Tollie was getting an inkling of the nature of this odd virtual world.

Ancient Egyptian looked at them hesitantly, deciding something in his mind. Evidently, Tollie and Mouse were friendly enough in manner that he went ahead and asked them for help with a problem he was having.

"Pharaoh sent me to find the height of the Great Pyramid and I cannot climb it to measure any of its sides. It is too smooth. Also, I cannot measure its height directly because its top is over the center. How can I find out how tall it is?"

Tollie looked up at the steep inclines of the Great Pyramid, noting how high it was and wondered how to find out just how high. "Well, Mr Mouse, do you know if it can be done?"

"Oh, yes, there is a way," Mouse assured him. "Remember what Carl said. If you believe there is an answer it will be possible for you to find it."

"But I can't climb that pyramid either."

"Maybe you don't have to climb it."

Mouse seemed to know everything. Tollie again found this irritating, but he had to give Mouse some leeway. He was a widely-traveled mouse and it was only his job.

"An indirect measure, you mean?"

"You're on the right track," Mouse encouraged him.

Tollie thought out loud. "What if there is a way to measure it indirectly?"

He thought and thought. He was becoming more relaxed about thinking through these new challenges. His face was not nearly as scrunched up as the last few times.

"You could measure something else that would give you an idea, something that you could measure directly? That would give you an idea of how high the Great Pyramid itself is."

"But what?" asked Mouse.

"Hmm..."

The sun was beginning to set and the air was cooling as the shadows grew longer. Tollie glanced at it, squinting. A light dawned on his face, the light of understanding, not the sun's rays, though they had something to do with it.

Mouse was smiling happily, humming his little tune, "Happy, happy are the mice..."

Tollie looked around. He found himself holding a yardstick, as he had thought that he would need one! Strange place, this "virtual" reality!

"Now you can measure, Tollie," Mouse hinted, though it seemed a somewhat trivial and obvious hint. He had a yardstick so he could measure. So what?

"But," Ancient Egyptian interjected, "what can he measure? Certainly not the height of the Great Pyramid." The poor fellow still had a worried look. Clearly, he did not want to disappoint Pharaoh and wanted to return with the required measurement.

Tollie patted his shoulder. "I have an idea that will help you."

Ancient Egyptian turned his profile to Tollie and smiled a smile that Tollie knew was a happy smile even though he could only see half of it.

"We are going to measure the shadow of this yardstick," he said. "When the shadow is the same length as the height of the yardstick, we will measure the shadow of the Great Pyramid, which will at that time also be the same length as the height of the pyramid!"

Ancient Egyptian considered this, then smiled, being careful to keep his profile turned to Tollie. "Yes, yes, indeed, that _would_ do, oh, yes, indeed!"

Tollie was gratified to see the worry lift from Ancient Egyptian's face.

Mouse danced to his own tune in the background. He was celebrating. Tollie had solved his first puzzle!

"Tomorrow, when it is early and the sun is at the right angle, I will measure it!" Ancient Egyptian announced. "I cannot thank you enough! Now I will be able to go back to Pharaoh with the measurement of the height of the Great Pyramid itself!"

Tollie smiled. He had solved the puzzle this time and it felt good. Ancient Egyptian was overjoyed. Mouse was still dancing to his little tune, but he winked at Tollie. It seemed to Tollie that Mouse was giving him due credit for solving Ancient Egyptian's dilemma. Tollie had to admit to himself, with a certain pride, that he _was_ getting better at this.

Ancient Egyptian thanked them and wandered off, to do his measurement the next day and return to Pharaoh, having accomplished his mission successfully.

### CHAPTER 6

THE GRUMPY CAMEL AND THOUSANDS OF BANANAS

Mouse tapped Tollie's shoulder. "See that camel trotting over? I think you will be facing a new challenge anon."

Tollie saw Camel stopping by a large pile of bananas that he had not noticed before.

"Hello," Camel said in a gruff voice, sounding something like a snort. His wide nostrils even flared.

"Hello, Mr Camel," Tollie replied formally, in case, like _Mr_ Mouse, Camel was a stickler for being properly and formally addressed. "I'm Tollie. You can just call me 'Tollie.'" He smiled and held out his hand, then realized that, unlike Mouse, who could shake hands with his hand-like paw (even as he stood and walked on only his two hind legs), this was just a regular four-footed talking camel. He looked grumpy, but Tollie dismissed this. He thought all camels looked grumpy.

Camel looked at the horizon. The desert stretching toward it went on endlessly. The vastness of the distance seemed to discourage Camel, for he snorted again, a very resigned sounding snort. "See these bananas?"

Tollie and Mouse both duly nodded.

"A thousand miles in that direction is an oasis." He indicated the westerly direction where the sun was now setting. "At that oasis are many hungry camels. They need all of the bananas I can get to them. But, I have to eat a banana for every mile I travel. I can only carry a thousand bananas at most. So, if I took a thousand bananas, I would get there with no bananas at all. They would all be used up to make the trip. Not a good thing. There are three thousand bananas in this pile. I can take a thousand and set out and I can drop some bananas at any point in between here and the oasis and come back to get more bananas. I can do this again and leave piles of bananas at different places in between. I need to get as many bananas as possible to the camels, they are _very_ hungry. How can I get the most bananas possible to them?"

Camel looked grumpier than before considering this.

"Tollie here is good at solving problems," Mouse put in at this point. Unlike his feelings toward cats, he had nothing against camels. He had nothing for them either, but at least nothing against them. "I'm sure he can help you."

Tollie was not so sure and wished Mouse had not volunteered his services. He had no idea how, starting with 3,000 bananas, taking only a 1,000 at a time, and eating 1 banana for each mile he traveled, Camel could get the most bananas across the desert to the oasis 1,000 miles away.

Mouse was just peering at him unhelpfully.

"Mr Mouse," Tollie implored, "you always know the answers. Don't you have even a hint?"

Mouse grinned. "Happy to help, Tollie. Think of leaving with 1,000 bananas, dropping some off somewhere in between, and returning to pick up another 1,000 bananas and setting out again, only to once more drop off some bananas somewhere in between. Remember, on the second trip you can use the bananas you dropped off on the first trip. Then you can return and set out on your third trip, taking the last 1,000 bananas from the start point. Count up the trips, Tollie. That's my hint to you."

It was a lengthy hint, but Tollie was grateful for any hope of getting a handle on Camel's problem. He felt sorry for Camel, as he seemed genuinely concerned about his fellow camels, hungry at their oasis. They had water, but no food!

Camel watched hopefully as Tollie began thinking. He knew Tollie was thinking because of the way his face was scrunched up. Tollie was getting used to thinking things through, but this was a tough one and he had reverted to scrunching up his face while thinking about it.

"One trip out, one back, one trip out again and one back again, then one out and that's the last outward-bound trip, and the last trip of all. Five trips total." Tollie pondered this for a while, having followed Mouse's hint.

Camel watched him expectantly. Mouse had a little smile on his face. "Think about how many trips starting at the second thousand bananas, too," he hinted.

Tollie had hardly begun thinking about the first trip, but now he took only a few moments to grasp the implication. "Three. You already made one trip out and one back, two trips out of five." As he said this, he began to see a possibility of getting an answer to how to get as many bananas across as possible.

The answer was there, he just had to puzzle it through.

"Five trips from the start, three trips after picking up the second bunch of bananas, only one trip out with the third, last bunch, five trips in all," he mused. "Maybe fifths and thirds..."

He thought on it. Mouse waited, an encouraging gleam in his little round eyes.

"All right, I have an idea." Tollie took a deep breath.

"Start out with the first 1,000 bananas. Go one-fifth of the way, or 200 miles, because you have four trips ahead of you still. You will have to pass this same 200-mile distance five times using up 1,000 bananas in all just for this part of the trip. And, you need to leave enough bananas so Camel... er, _Mr_ Camel... has one banana to eat for every mile he travels. Soooo... he can go out to the 200-mile point, which is 1/5 of the 1,000 mile distance and leave 600 bananas, eating 200 going out and 200 going back, using up only 400 of the first 1,000 bananas."

Mouse was doing a bit of a dance and humming quietly in the background, almost to himself, knowing Tollie was doing his thinking right. This encouraged Tollie to go on.

"Now, Mr Camel sets out from the start point for the second time, with the second thousand bananas, leaving behind the last thousand bananas. He has three trips ahead of him and 600 bananas waiting for him at the 200-mile point. When he gets to the 200-mile point he has 800 bananas since he had to eat 200 bananas to get there. He picks up 200 bananas, giving him his maximum load of 1,000 bananas again. That leaves 400 bananas at the 200-mile point."

Camel seemed to hearten, seeing that there was some chance here of getting the most bananas he could to his fellows at the oasis. Tollie went on.

"This is his third trip, an outward trip, the fourth trip will be back to the start point and the fifth trip will be the last one, outward. This means he can go 1/3 of 1,000 miles further, or 333 miles, taking him to the 533-mile point, where he can leave 334 bananas and still have 333 bananas left to get him back to the 200-mile point, where he can pick up 200 bananas to get him back to start point. This still leaves 200 bananas at the 200-mile point. It also leaves 334 bananas at the 533-mile point. He will use these bananas for his last trip out to the oasis."

Camel snorted, but it almost seemed an appreciative snort, which, no doubt, is what it was. It is hard to understand the emotions of camels. Tollie continued, coming to the last leg of the journey. Camel paid careful attention.

"Okay. Now he sets out on the last trip, with 1,000 bananas. He will eat 200 to get to the 200-mile point, but he will pick up the last 200 bananas there, and have 1,000 bananas again!"

Tollie paused to breath. This was, he thought, the toughest challenge he had so far faced.

"With a thousand bananas at the 200-mile point, he goes to the 533-mile point, eating 333 bananas to get there. BUT, he picks up the 334 bananas he left there on the second trip. So, he has 1,001 bananas now. He can only carry 1,000 bananas at a time, maximum load. So, he eats one banana for the first mile _before_ he sets out. He then takes 1,000 bananas, eating one per mile.

"He has 467 miles to go to the oasis and his fellow hungry camels. This means he will eat 467 of the 1,001 bananas, leaving him 534 bananas when he finally arrives!"

Tollie let out his breath. He had concentrated deeply to get this result.

"Mr Camel, I think 534 is the most bananas you can get across to them. I hope that will be enough?"

Camel thought this over. "Yes, yes," he said at last. "I believe you have it right, and, yes, 534 bananas will be enough. My friends will only have to travel 100 miles to the coast, where they can get the food they need. There will be five of us, including myself, naturally. So, there should be more than enough! We could even eat a few before we set out. I don't know how to thank you."

"A camel in need is a camel indeed," Tollie said graciously.

"Well, I better get going," Camel announced a little apologetically. "They are waiting for me." He trotted off, a swagger in his walk. He knew he would be a hero among his fellow camels when he got to the oasis.

Mouse and Tollie watched him go. He would at least be traveling in the evening, so it would be cooler for him. Tollie felt good. He had been able to help. Mouse said, "A good deed nicely done. You are learning and growing here, Tollie."

"I think you're right, Mr Mouse. I really was glad to help Ancient Egyptian and Camel. I like that I could find the answers they needed. I think I learned that from Carl. He was quite a kid, wasn't he?"

"Oh, yes, I would say so," Mouse said, with a wry smile.

For the first time since he had entered this brave new virtual world, the full sense of its wonders and possibilities hit Tollie. It seemed he could just think of anything and Mouse could get him there. So many choices, so many things to do, so many places to go. Where to now? He was not feeling the least bit tired, though it felt to him that they had done a lot of traveling together so far. Time, too, passed differently here. The little clock in the corner showed only about half an hour had passed, though it seemed quite a bit longer to Tollie, in spite of not feeling tired as he surely would in the "real" world. This was a very different place.

Unconsciously, he rested his hand on Mouse's back. He was becoming used to this, comfortable with it, it seemed the only natural thing to do. His reflections on time, however, led Mouse to transport him to a kitchen, of all places. Well, where else would one use a timer?

### CHAPTER 7

TWO TIMERS – THE HARD-BOILED EGG STORY

An obviously inexperienced young chef, no doubt an apprentice, was looking very worried. The reason soon became clear as he gestured toward two egg timers on the counter, the kind that measures a specific amount of time by allowing a certain amount of sand to pass through the narrow middle neck of an hourglass container.

The egg timers were clearly labeled "7 minutes" and "11 minutes."

"But," Young Chef groaned, "I have to hard-boil an egg properly, which means boiling it for exactly 15 minutes, to get it just right. There is no 15-minute timer!"

"Let's think about this methodically," Tollie said with a newly found confidence. "There might be a way to do it with the timers you have."

Mouse remained silent, but Tollie knew his body language well enough by now to know that Mouse knew there was indeed a way.

"Please," Young Chef implored him, "if you can help me I would be ever so grateful.'

Mouse took a seat and listened. It was a tough puzzle but this time he offered no hint. He hadn't been asked for one. He thought Tollie could solve this one without any hints.

"Something to do with the difference between 7 minutes and 11 minutes." Tollie did his thinking out loud. ""Not the simple addition because 7 plus eleven is 18, too many minutes. But, then, 11 minus 7 is 4, too few minutes. Of course, 4 plus 11 is 15, the right number of minutes to hard-boil the egg perfectly."

Young Chef listened to him eagerly. "Yes, somehow we must get 11 and then 4 minutes. We have the 11 simply enough from the timer, but how to get the 4 additional minutes?"

Tollie thought in silence for a while. Young Chef waited. Mouse sat relaxing, not looking worried at all, at all. This did not irritate Tollie this time, since he believed Mouse was relaxed because he thought Tollie would solve this challenge, as well as he had now solved two in a row. Mouse believed Tollie was a good learner. Carl Friedrich had hinted at something of the sort as well. So, he applied their advice now. The starting point was to believe there was an answer.

"Let's try turning them both over, starting them both off at the same time." Tollie paused, his face scrunching up just a little. Then he resolved something in his mind and continued.

"When the 7-minute timer runs out, turn it over again!" Here was the key he had figured out. "Now, when the 11-minute timer runs out, the egg has cooked for 11 minutes total. We need 4 more minutes. There are 4 minutes on the _bottom_ of the 7-minute timer. So, turn it over yet again, putting that 4 minutes on the top. When that runs down, the egg will have cooked for 15 minutes and it will be a perfect hard-boiled egg."

Young Chef was overjoyed. "That will work! Thank you, thank you. Come, come," he said. "You two deserve a few moments of rest. I can see you would like to sit and stretch your legs for a while."

He led them through the swinging doors and down a long corridor, emerging in the lobby of the hotel to which belonged Young Chef's kitchen. The concierge at the desk greeted them cordially. "Of course, you are welcome at the Hotel Frodlaw. Have a seat for a while. Those two easy chairs by the window should serve you well for some relaxation."

Tollie noted the sign "Hotel Frodlaw" and thought it an odd name for a hotel. But it did seem oddly familiar...

Young Chef took his leave. "I have to cook the egg," he told them, "but I will return. Enjoy your rest."

As they rested, they watched as a bustling trio entered the lobby from the other end. Their friends the merry triplets had arrived, hipping and hopping along! Tollie chuckled and Mouse allowed himself a small smile.

### CHAPTER 8

THE CASE OF THE MISSING DOLLAR

Hop Hop, Hip Hip and Hip Hop made their way to the desk. Hip Hop beseeched the concierge, "Sir, we have very little money. Can we share a room for the night to defray the cost?"

"Yes, that will be quite all right," the concierge said. "It will be $30. Will that be all right?"

With much ado about it, the triplets went through their pockets, pulling out even their change to assemble the right amount. Their pleasure was great when they finally got $30 together and gave it to the concierge, who put it in the register and signaled the bellboy over to help the triplets with their luggage.

"Oh, no," Hop Hop protested, "we have no money left for a tip."

"It will be my pleasure," the bellboy assured them. "We all fall upon hard times now and again and you are nice enough men that your good will is quite enough." And he took their bags and led them up the stairs to their room.

"Nice hotel, this," Tollie observed.

"Yes, the Frodlaw has always had a good reputation for its service, even though sometimes they get things backwards."

Mouse stretched and yawned, a quite curious sight, as Tollie had never seen a Mouse yawn. Mouse even covered up the yawn with the plaid sleeve of his jacket.

When the bellboy returned, the concierge summoned him and handed him $5. "I have just overcharged those men for their room. It should have been $25. Please take them the $5 back." He handed over five single dollar bills, and the bellboy left to return it to its proper owners.

After a while, Hop Hop, Hip Hip and Hip Hop came downstairs, deep in conversation, evidently confused about something. Hip Hip was saying, "We got back $5 and then we tipped the bellboy $2. So, we got back $3 overall, meaning we each paid $9 instead of $10 for the room. For three of us, that means 3 times 9, which equals 27. Then we gave $2 to the bellboy for his tip. That makes $29. What happened to the extra dollar?"

"It is true!" Hop Hop agreed. "We each paid $9 and 3 times 9 is 27, and you add the $2 we gave to the bellboy and get $29. There _is_ a missing dollar! How can that be? What happened to it?"

They then noticed Tollie and Mouse for the first time and hipped and hopped over to greet them, grinning merrily, and, of course, presenting the problem of what they had paid for the room and why there was a missing dollar.

Tollie was perplexed, and even Mouse looked uncertain. The logic was there and the multiplication and addition seemed correct. This was not easy, for, however he looked at it, Tollie could not "find" the missing dollar or explain how it added up to only 29 when it should have added up to 30.

The concierge, having overheard the conversation, came over to help. "This has happened at the Frodlaw before," he said. "It took us a while but we were able to see the source of the confusion. It is in how Mr Hip and Mr Hop were stating the transactions."

Tollie, Mouse and Hop Hop, Hip Hip and Hip Hop were all paying close attention. They truly wanted this explained to them, as the missing dollar was bothersome.

"It is true that each of the triplets paid $9, making their expenditure a total of $27. They also paid $2 to the bellboy, but they made the mistake of adding that to the $27, when they should be subtracting it from $27. They were supposed to have paid $25. Add to that the $2 tip for the bellboy and you arrive at the $27 they actually paid. Put another way, they paid a total of $27. Subtract from that $27 the $2 tip and you arrive at $25, the cost of the room itself. There is no addition of $2 to $27. The $2 needed to be _subtracted_ from the $27!"

"I can see it now," Mouse said. "It is a matter of how they were saying it that led to the confusion."

"Yes," agreed Tollie. "It was a confusing of doing addition when it was subtraction that was called for."

Hip Hip, Hop Hop and Hip Hop giggled, embarrassed at their having made a mess of their calculations.

"It is not a problem," the concierge assured them. "Several of our guests have done the same. Of course, the first time it happened it was difficult to figure it all out. And, I understand it is my mistake in the first place, for having overcharged you"

"Oh, that is quite all right," Hip Hop assured him in return. "You were honest enough to return the money, and we got to tip the bellboy after all, so all is well."

"I am glad to have been of service in clearing up the mystery of the missing dollar," the concierge told them with a bow, and returned to the hotel desk.

"Well, we must repair to our room and unpack," the triplets told Tollie and Mouse, speaking in unison as they tended to do, unless, of course, they were confused about something that they needed to discuss among themselves. "Perhaps we shall see you later. In fact, I am sure we will!"

Tollie waved at them, smiling, and Mouse made a convincing "wave" with his "hand." The triplets jaunted up the stairs, bringing much mirth to all who witnessed them. The ways they moved were funny to behold.

When they were gone only moments, Young Chef returned. "I have a few things to show you. Come with me. You are going to enjoy this. I hope you like apples. And oranges."

Tollie and Mouse indeed hoped there was food involved. They were hungry and followed willingly. Tollie liked apples. So did Mouse, but he had never eaten an orange, so he could not compare apples with oranges the way Tollie could. Of course, Tollie would eat oranges sometimes, because they were good for you, but they were not as tasty to him as apples. Mouse would have to learn.

### CHAPTER 9

APPLES AND ORANGES

Through the swinging doors again and down the long corridor in the other direction now, Young Chef took a right turn and opened the first door, beckoning them with a flourish to enter.

In the center of the room were three boxes, all with labels: APPLES, ORANGES, and APPLES AND ORANGES.

"At least we know where the apples are," Tollie observed.

"Not quite," Young Chef said. "You see, all of the boxes have been given the wrong label. You can have all the apples you want, or oranges if you wish, but you must first find the right labels for all of the boxes."

"That's crazy," Tollie objected. "Who would deliberately put wrong labels on them?"

Young Chef shrugged. "It's a crazy world, what can I say? It might be someone's idea of a joke. Or, perhaps it was not done deliberately."

"Indeed," Mouse said enigmatically. Tollie scrunched up his face. "Crazy indeed," he said, rolling his eyes heavenward.

"You have to follow the rules," Young Chef warned them sternly. "You can take fruit out of the boxes, but you must take the minimum number of fruit and you must take them from the minimum number of boxes. Proceeding this way, you must correctly label the boxes with the minimum number of removals from the boxes. Only if you label the boxes correctly can you have your snack of them."

"Humph!" Tollie snorted. "Silly, but let me work on it.

His face relaxed as he pondered. The minimum number of fruit from the minimum number of boxes. He would have to reason his way to the answer. He _believed_ now that there was an answer!

"If you took one fruit out of APPLES, one out of ORANGES, and all of the fruit out of APPLES AND ORANGES, you would know the answer and could label the boxes correctly. But that seems excessive. The problem calls for a simpler answer, only because it has been presented as a problem, and you would not do that if there were not a simpler answer than just using 'brute force' to get them labeled right.

"So, let us look at the unique case. The box labeled APPLES AND ORANGES is the only one with a dual label. The other boxes name only one fruit. Let's pull a fruit out of APPLES AND ORANGES."

Tollie reached in and pulled out an orange.

"Now, this box must contain oranges, as I have here an orange and the box cannot contain apples and oranges, for it is mislabeled. It has an orange in it and so it cannot be APPLES. Again, it cannot be APPLES AND ORANGES because that label is wrong. Let us therefore label it ORANGES now, and we have labeled one of the boxes correctly."

Young Chef and Mouse nodded in agreement.

Tollie continued, "The box that is labeled APPLES cannot contain apples because the label is wrong. And it cannot contain only oranges because we now know where the oranges are. So it must be labeled APPLES AND ORANGES, the only option remaining. And, that leaves the box labeled ORANGES, which must contain apples only as that is the only label left, and we must label it APPLES.

"If you had pulled out an apple instead, you would just reverse the procedure."

"Let's see," Young Chef said, switching the labels according to Tollie's answer and then opening the boxes to find them correctly labeled as Tollie had indicated. "Apple, anyone?" he asked.

Tollie took an apple and thanked Young Chef, who then offered one to Mouse. "Here, Mr Mouse, have one yourself."

Mouse took the apple and slipped it into a large pocket in his ridiculous plaid jacket. Tollie failed to completely stifle a laugh. Mouse looked a bit offended but it quickly passed. "I am saving it for later, when I hope I will have some cheese to eat with it. For now, I would like to try an orange. There is always the opportunity to be considered to try something new."

"Cheese, eh?" Young Chef mused. "There is a possibility that we could get some cheese for you, but, here, have yourself an orange for now."

Mouse looked at the orange dubiously. He could not see just biting into it like you would an apple.

"Here," Young Chef offered, "let me peel that for you."

He did so and handed mouse a wedge of orange. Mouse nibbled at it and drew in his cheeks. "That is potent," he said. But he ate it and another, and shared the rest with Young Chef and Tollie. Tollie was hungry enough by now that he had eaten the apple and was glad for some of the orange as well.

"You mentioned a possibility of cheese, Young Chef?" Mouse asked eagerly.

"Oh, yes," Young Chef responded quickly, "I am having a little problem with our cheese machines and if you can help, you can both have some cheese. Tollie, save an apple for yourself, you can have it with cheese. Mr Mouse, I know you have your apple ready for cheese."

"Cheese!!" Mouse said with relish. He loved cheese, as mice will.

### CHAPTER 10

THE BIG CHEESE

Young Chef led Tollie and Mouse back to the long hallway, and took them to the furthest door, at the end, facing them. He solemnly opened it with an impressive flourish. They entered a fully working cheese factory, which to Mouse was about the most wonderful thing in the world.

The pungent odor of cheese set Mouse aquiver. His eyes gleamed with excitement and he wriggled his nose animatedly, his entire being suffused with sheer anticipation. He could barely contain himself.

Ten machines along one wall clanged, whirred and made strange hissing and slicing noises, spitting out wedges of cheese onto a conveyor belt. Tollie thought one of them would last him a whole day, so large were the wedges. Mouse looked on, struggling to maintain his customary patient composure. "How much does one of those wedges weigh?" he asked, trying almost successfully to maintain a calm exterior.

Young Chef said, "Funny you should ask. They are supposed to weigh one pound each."

"Big hunk of cheese," Mouse observed. "But I certainly would like one!"

"You can both have one but you must first help me with a little problem we are having."

Tollie sighed. Ever since entering the World-Wide Web, this blooming buzzing virtual reality, he had been faced with one problem after another. By now he could tell when yet one more was coming up. But Mouse waited eagerly to get on with it. Even if they had to earn their cheese, he wanted to move things along.

"You see, one of these machines is making a slightly heavy wedge of cheese. One ounce too heavy, to be precise. One tenth of our cheese wedges weigh one pound and one ounce instead of the required pound they are supposed to weigh. You can have a couple of the heavy ones, but first we need to know which machine is making the heavy wedge."

"And you can't tell just by looking, I suppose," Tollie said, already knowing the answer.

"No, they all look the same. But we do have a scale."

"Then we can just weigh the wedges coming from each machine, taking ten weighings at most, and find out which machine is making heavy wedges, can't we?" Tollie asked, still harboring a feeling that this would be too simple to be a possible answer – for, then, what would be the problem?

Young Chef pointed to a truly huge scale in the corner. "It has a large pan you can put the cheese wedges on and it has a digital readout, giving the weight in pounds and ounces. Our problem is that it will shut off after one more weighing, and then we will never be able to turn it on again. Some malicious hacker got into its inner computer and put in a swatch of computer code that causes it do shut down forever after a certain number of weighings. We cannot fix this, we will have to send the machine back to the manufacturer. But we would like to find out which cheese machine is making the heavy wedges of cheese before we send the scale away. We are hoping to do this using up the one weighing we have remaining."

"Could it not be that this same malicious hacker also caused one of the cheese machines to put out heavy wedges of cheese, just to mess up your output?" Tollie asked.

"It is possible," Young Chef admitted. "Frodlaw Cheese is famous the world over for its tart flavor and consistent quality. It used to be made only for the hotel, but the demand for it grew so great that we enlarged the factory to accommodate the cheese lovers. We need to maintain our reputation for consistency and our wedges need to all be one pound, so we can package them uniformly and ship them out. So, the sooner we can identify the problem machine, the better. I just hope there is a way to do it with just the one weighing."

"That's a tough one!" Tollie exclaimed. It seemed on the surface of it to be the hardest challenge yet posed to them.

Mouse was all ears, even as his nose still quivered at the delectable cheese aroma he could not resist. Tollie needed a hint, and that was his job. He held his paws together in front of him, mentally composing a good hint. It should not give away the answer, but it should help Tollie come to one. "That scale," Mouse said, indicating it with a broad sweep of his right paw, "gives you a reading in pounds and ounces. If you took one wedge from the first machine, it would give you a reading of one pound, or one pound and one ounce, depending on whether or not the first machine is the defective one. You can put any number of cheese wedges on the scale. You can see it is large enough to hold many of them."

Mouse stopped. This was it, the whole of his hint. Tollie's face must have shown his disappointment, because Mouse added apologetically, "It's a good hint if you can follow it forward."

Tollie accepted this, as Mouse had never led him astray. So he thought on it.

And thought on it over and over, though no solution seemed to be presenting itself. Young Chef and Mouse were looking everywhere except at Tollie, politely giving him time for his reflections.

"One pound and one ounce," Tollie thought out loud. Then he fell silent again, still not getting past this to any broader solution that would take into account all of the machines. "If the first machine is not the heavy machine the scale would read one pound and zero ounces. If it was the heavy machine, the scale would read one pound and one ounce..."

After a long while, Mouse relented. "Ten machines. You can take all the wedges you want from each of the machines..." He let the suggestion linger in the ensuing silence.

Tollie had heard him, of course, but he was still not seeing how it would help him in any way. "Any number of wedges from any of the machines, or even all of the machines... Pile them up on the scale... Read the total weight in pounds and ounces... Then identify the heavy machine..."

It came to him in a flash, like so many of these new challenges. He had read about that somewhere, he remembered. It was called the "Aha!" reaction.

He composed himself at once, for he had the solution now in his mind.

"Take one wedge from the first machine, two from the second, three from the third, and so on until you have taken ten wedges from the tenth machine. Put all of these on the scale and take your one reading."

Young Chef looked at him quizzically. Mouse was starting to get excited. Tollie, he knew, had the answer, and soon he, Mouse, would have cheese! It was like being in Heaven.

Tollie explained. "Your reading will be in pounds _and_ ounces. If you took one wedge from the first machine and two from the second, and one of those two machines was the heavy machine, you would get a reading of either three pounds one ounce, if the first machine was the heavy one, _or_ three pounds two ounces, if the second machine was the heavy one. This would be because you had one wedge from the first and two wedges from the second machine. You can follow this through to all ten machines and use the same logic – extrapolation. Say, you had on the scale ten wedges from the tenth machine and it was the heavy machine. The reading would be a number of whole pounds _and_ ten ounces more. The number of ounces over the even pound weight will tell you the number of the heavy machine."

Young Chef slowly came around. "Yes, it would..."

Mouse coughed discreetly.

"Oh, yes, it would, of course. And I must thank you. And, I do owe you each a wedge of the cheese, and I will make sure it is one of the heavy ones, just as soon as I carry out the weighing."

Mouse looked a bit put off. He would have been happy to just have a wedge of cheese right now, but he saw that Young Chef wanted to be generous, and he would, after all, end up with an extra ounce of cheese!

While Young Chef was performing his weighing, Tollie noted to Mouse, "We have been here a long time. It seems like we were just led along from one thing to the next ever since we met Young Chef."

"Oh, of course," Mouse said, "this is what is called 'surfing the net.' Sometimes you just find yourself being carried along by the links that hold this world together, and you can find many new things that you would not have thought of if you did not pursue your natural curiosity."

Tollie considered and felt he had, indeed, enjoyed the random encounters as they had provided him that pleasure that can derive from the unexpected.

"But," Mouse continued, a bit of warning in his tone, "you will have to be careful where this curiosity takes you or you can end up in a dangerous place. There are all kinds of things here and all kinds of people and creatures. They are not all safe. Just be careful. So far it has been all right, because Young Chef is an honorable man."

"He is that," Tollie agreed. "And I can see that there would be dangers in this world, just as in that 'real' world I come from. You are a creature of this world, so you understand it better, but it is understandable anyway, because of the similarity between the worlds."

Mouse smiled enigmatically, wishing it were this simple. But he had faith that Tollie had natural common sense in addition to his natural curiosity.

Tollie felt it was time to go and placed his hand on Mouse's back. A helicopter appeared and landed next to them.

"You are figuring out one thing, aren't you, Tollie?" Mouse suggested.

"What's that?" Tollie asked.

"Oh, the little matter of who is in charge here," Mouse said, an enigmatic smile playing across his face. "You are, of course."

"Let's get aboard," Tollie said eagerly, because he had always wanted to ride in a helicopter and here was his chance. It was a virtual helicopter, of course, but it was still exciting. For the moment he did not realize that he was indeed in charge, but Mouse realized it and was happy for Tollie. He went along and clambered aboard the helicopter, his chunk of cheese in hand. Whoever was in charge, Mouse had his cheese!

### CHAPTER 11

THE FRIENDLY ISLE OF AMITI

The pilot welcomed them aboard. "I'm Madison, but you can call me Maddie," she introduced herself. "It's good to have you, just take your seats and fasten your seatbelts. We are going over the ocean, just as Tollie wanted."

Mouse just nibbled at his cheese, while Tollie wondered at the seeming miracle that Maddie knew just what he wanted.

They were off!

In the distance lay a small, long island, and as they got closer, they could see that there was a commotion there, trouble.

"That is the island of Amiti," Maddie told them. "It is known for the friendliness of its people, the Amitians, a people small in size, but large in generosity and kindness. They are also called the Friendly Islanders. They always welcome strangers with arms outstretched and go out of their way to accommodate the needs of their visitors. So, if they are in trouble, we should help them."

"Let's get closer and take a look," Tollie said.

Mouse put aside his hunk of cheese for the moment, to pay attention to what was transpiring on the Friendly Isle of Amiti.

As they got closer, they could see just what the trouble was. The islanders were gathered together at one end of the island, while a large fire moved toward them from the other end of the island, having already burned down their village. The islanders had left only the possessions they had been able to carry.

Their king, Iti Biti Amiti the Fifty-Fourth, stood tall (at 5' 2 ½") among his people, seeking counsel and a way out of their dilemma. The island was ten miles long and the fire was approaching them, at a speed of about five miles an hour as it happened, as the prevailing winds were this speed and heading in their direction. King Iti Biti Amiti was trying to figure out a way to save his people from being consumed in flames as the fire inexorably approached. They did not have much time.

Tollie took this all in as they approached. "We must land next to them," he told Maddie. She took the helicopter down and they disembarked to join the Amitians and their king.

Even though they were in dire straits, the Amitians welcomed them with their customary warmth. "Welcome to Amiti," King Iti Biti greeted them. "We would love to invite you to share a meal with us, but, as you can see, we are having a problem here. Maybe you can help us?"

"Of course, anything we can do, we will do," Tollie answered, and Mouse and Maddie nodded in agreement. "We could just take off in our helicopter, but we could not take your people with us, so we must come up with a solution and that fire is approaching quickly!"

"I know you cannot take us with you, but what can we do? Whatever it is, we must come up with a plan fast."

Tollie thought on this. Mouse looked on and did not seem to have a clue for him this time. Maddie was concentrating, looking at the raging fire despairingly. How could the Amitians be saved?

"The fire is coming on strongly," Tollie said. "We must somehow stop it and stop it before it reaches the people of Amiti. It is about six miles away now, a little more than an hour away, and the edge of your island, the far coast, is about two miles in the other direction. You cannot just run from it. That would at most buy you a little time, but it would not save you. You would run out of space to run."

"That is correct," the now even more forlorn Iti Biti Amiti acknowledged. "At this point I just want to save our people." The islanders all looked morose as their King did not seem to know what to do and none of them were seeing a way out either. "If we can survive this fire, we should be all right. Dry season will soon be over and we can rebuild the village and re-grow our food supply, and there are plenty of fish we can eat in the meantime."

Which prompted a thought in Tollie's mind. "Why can't you just go in the water?" Tollie ventured, giving what seemed an obvious solution.

"Ah, would that we could," Iti Biti replied. "But it is unsafe. At this end of the island the ocean is full of sea beasts that would simply destroy us in the water. We can swim and fish at the other end of the island. That is why we built our village there. For some reason the sea beasts do not go there, so it is safe for us to swim near the village.

"Or, where the village was..." he corrected himself. "But the fire prevents us from getting into the waters there. It is too late."

Tollie wanted to ask about these sea beasts, but realized this was not the time for his curiosity to reign, but, rather, time for him to rein it in! They were probably sharks, giant squid, or octopi and such, he reasoned. It did not matter, anyway, for if the Amitians believed in these "beasts" they were as real as anything else to them, and nothing could get them to enter the waters here.

Tollie glanced into the distance, rapidly closing, at the coming fire. It seemed nothing could stop it as it consumed everything in its path. The thought of the fire "consuming" everything stuck in his mind, and he ran it over and over in his considerations.

Of course! A fire could not burn if it had nothing to consume. It had to have fuel. Remove the fuel and the fire would die.

"I have an idea," he addressed King Iti Biti Amiti.

Iti Biti regarded him gravely. "Whatever it is, we need some answer and a realistic course of action."

"We must set fire to the island from here to the far coast, which is about two miles away. We set the fire and then follow it to the edge of the island and wait there. When the big fire reaches the land that has already been burned, it will go out, since it will have nothing to burn for fuel. The land and everything on it that is flammable will have already been used up by our little fire."

Iti Biti nodded. "Yes, that would work. The entire island would still be burned up, but all of us would survive."

He turned to his people, drawing himself up to his full height, and explained to them what must be done. They listened, doubtful at first, but then understanding that setting another fire would, oddly enough, save them from the approaching fire they now faced. They set about gathering kindling to give their new fire a good start.

Iti Biti watched them, hope showing in his expression for the first time. "I must thank you, Tollie. You certainly came up with what seems the only solution to our problem."

Mouse chimed in. "Tollie has become quite good at solving different problems. It is his challenge here in our world-wide web and he has been rising to the occasion each time he has encountered a new challenge. He has learned much since I first met him."

"A good thing for us," Iti Biti said, watching as his people set the smaller fire that was to burn the land between them and the coast.

The new fire took, and burned in the direction of the wind, burning the two miles between them and the other end of the island as the original fire approached from their other side. The two miles burned up after some minutes and the islanders began to run over the burned land, away from the big fire. They would be safe now, since the big fire would stop here, having reached the scorched land and having nothing to fuel it.

"I must take my leave of you and go with my people now," Iti Biti said. "Thank you for your help! But, you must now go, too. Come back in a year and we will have a wonderful feast for you and your friends!"

"Thank you, we surely will," Tollie said. "Good luck!"

King Iti Biti joined his people heading away from the fire and escaping to the safety of the island's end. Tollie, Mouse and Maddie climbed into the helicopter and lifted off. Amiti grew small in the increasing distance as they flew away, but they saw the big fire reach the burned land and expire in a matter of minutes, no longer a danger to the Amitians, who, huddled at the end of the island, watched gratefully as the last flickers of flame died away.

They waved at the Amitians from their distance and saw them wave back before the helicopter traveled too far away to see the island any more.

"A nice people," Tollie observed. "I hope they will be all right."

"They will be fine, Tollie," Maddie told him. They will fish for food until the wet season comes and then they will rebuild. Their lives are simple but comfortable and they will rebuild their village happily. It will take some time. After all, the plant life must return and grow enough for them to get material."

Mouse pulled out his cheese now and nibbled at the remaining hunk, which, Tollie noticed, was much smaller now than the original. Mouse had been nibbling while he had not noticed.

Seeing him eye the cheese, Mouse explained, "I was nervous for the Amitians and when I am nervous, I nibble cheese. It comforts me."

"Where to?" Maddie asked. Mouse looked at Tollie expectantly.

### CHAPTER 12

THE TRUTH OF THE MATTER

Tollie took a bite of his own cheese and some apple to chase it down. "Let's go to another island. I like islands."

"Island it will be then," Maddie announced. "See that one in the distance?"

Tollie and Mouse squinted. They saw an island about the size of Amiti, but holding two villages. It was more lush than Amiti at this time of year and had much plant and tree growth and more mountainous region. They could see the two villages clearly now. They were of roughly equal size and well-built, with stone cottages and cobblestone-paved streets. Small villages but well laid out, with clear main thoroughfares and a number of offshoot streets dotted with cozy houses.

"This is the True-False Island," Maddie told them. "It is so called because it has two villages on it and the inhabitants of those villages are quite opposite to each other. In one village, only the truth is told, while in the other only lies are told. Otherwise, you cannot tell the villagers apart, different as they may be in their capacities for truth-telling. I will take you down and you can see for yourselves."

She landed the helicopter delicately, finding an open flat area where they could step off easily. When the rotors calmed, the three disembarked. "You explore," Maddie said. "I will have to stay with the helicopter."

"We'll just take a short tour," Tollie said. "We have to see one of the villages – the Village of Truth, I would think. That would be better for us. Who knows how calamitously we could be misled in the Village of Lies?"

Mouse set out with Tollie. "I know of this island. It is not that easy to get to the Village of Truth, since we must first figure out which one it is."

"Hmm... I didn't think of that. How would we know which one is which? We couldn't just ask, for we would not know whether we were speaking to a Truth-teller or a Liar."

"Think of the Apples and Oranges problem we faced with Young Chef," Mouse suggested. "Since the labels referred to their respective boxes and since they were all wrong, we could figure out which was which by pulling out only one fruit. Perhaps there will be a question that would let us know which way to go to get to the Village of Truth, no matter which villager we encounter, whether he be Truth-teller or Liar."

Tollie looked doubtful.

Sure enough, they came upon a fork in the road, and there, where the path diverged into two, stood a villager. There was nothing in his appearance or demeanor to betray his status as a truth-teller or liar. You simply could not tell.

"Suppose," Mouse said, "that all people from Crete are liars. You meet a Cretan and he says to you 'I am lying.' Is he telling the truth or is he lying?"

Tollie thought this over, his face scrunching up a bit. "How can I answer that? If he is telling the truth, then he is lying. And, if he is lying, then he is telling the truth, and if he is telling the truth..."

"Yes, exactly," Mouse said, peering at Tollie through his bright eyes. "It is known as the self-referential paradox. The Cretan is referring to himself, but he is not simply describing himself, he is also describing his own statement within his statement. That is, the statement is about the statement itself. In a similar manner, if you can get the villager to have to answer self-referentially, you might be able to get the correct answer, no matter which villager it is."

The villager addressed them at this point.

"Hello there. You seem strangers here. Can I help you?"

The villager was friendly enough, but Tollie did not see how he could help them if they did not know which he was, truth-teller or liar. How could you ask him anything and be sure of getting a helpful answer?

"In the case of the boxes of oranges and apples," Tollie mused, "since all the labels were wrong, there was one thing you knew – that whatever the label on the box said, that was NOT what was in the box."

"True, true," Mouse said encouragingly. "But in this case, IF it is a truth-teller, then he will be saying which way the village is."

"And the liar would tell you the way to the wrong village, the Village of Lies, say, since we seek the Village of Truth..." Tollie thought out loud.

The actual villager watched them curiously. Tollie looked back at him. "I could just ask you if you are a Liar or a Truth-teller," he said, "but, then, if you were a Liar you would still say you were a Truth-teller, and I still would not know which you are."

The Villager smiled apologetically. Not very helpful but not harmful either. Tollie realized he would have to outwit the Villager, truth-teller or liar, either way. And he would have to do it in one question, as he already realized. If it could not be done in one question, it could not be done in two! So far, he had reasoned correctly, but he needed the one question to be the right one, the one that would work. He had simplified the problem, but he had yet to come up with that one question.

Something struck him about what he had said before, about asking the villager directly if he was a truth-teller. The truth-teller would say "yes" but so would the liar. You would get the same answer from both, but, even so, you would not know the truth. Was there a question about the road to the village that would get the same answer from both the truth-teller and the liar AND still tell you which road to take?

If you asked "Is this the road to the village of the truth-tellers?" and it was the road, the truth-teller would say "yes" and the liar would say "no." BUT, if it was instead the road to the village of the liars, the truth-teller would say "no" while the liar would say "yes." You don't know which villager you are speaking with, so the answers give you no information. A better question was needed.

Tollie pondered on this. Mouse looked down each road curiously. The villager stood expectantly waiting. Then Tollie asked the question.

"Is this the road to your village?" He indicated the left road.

Of course, if it was the road to the truth-teller's village, the truth-teller would say "yes (it is the road to my village)." And, as it was not the road to the village of the liars, the liar would also have to say "yes (it is the road to my village)." Thus, a "yes" answer would tell you the road led to the Village of Truth.

Likewise, if it was the road to the liar's village, the truth-teller would say "no (it is not the road to my village)." The liar, too, would say "no (it is not the road to my village)." Because it was the road to the liar's village, he would be forced to lie and say it was not the road to his village. A "no" answer would tell you it was the road to the Village of Lies.

"Well," Mouse said, "that question will do it!" Then Mouse chuckled quietly, and Tollie looked at him quizzically.

Oh," Mouse went on, "I was just thinking about the time I was the guide for an anthropologist. He had quite a different question he used to find out which road led to the village." And he laughed at his recollection.

"What other question could there be?" Tollie asked, doubting there was any other question that would do the trick.

Mouse laughed merrily. "He asked, 'Is it true they are serving free beer in the Village of Truth?'" Then he simply followed the villager.

Tollie laughed with Mouse and they set out for the Village of Truth. Of their adventures there, we will hear in a future volume, but, suffice it to say, they learned much and had a wonderful time. And when they were done, they returned to Maddie and her helicopter.

### CHAPTER 13

BETWEEN THE LIONS

Upon seeing them, Maddie did a little dance and sang, in the same tune as Mouse's little song:

"I fly, fly a whirl ybird,

Fly a whirlybird, bird,

Whirlybird..."

Mouse did his little jig and joined in.

"Happy, happy are the mice,

happy are the nice, nice,

nice, nice mice..."

Tollie wondered if everyone here sang the same tune! He recalled Ancient Egyptian's song. But, thinking there was nothing better to do at the moment, he joined in with the fun.

"We are, living in a dream,

living in a dream, dream,

streaming dream..."

And the three of them danced and sang happily together, forming an improbable but merry trio in a circle under the beamish sun.

When they were aloft in their "whirlybird" they looked back at the island, the two villages and the expanse of ocean that grew between, until the island was but a speck on the horizon.

Maddie called their attention forward, to see a little island full of grass, lions and one sheep, a curious place. "This island is populated by 100 lions, one sheep and much grass," Maddie told them.

Mouse looked and visibly shuddered. "Those are cats, Tollie, nothing but big cats!"

"We are safe, Mr Mouse, we are up here and they are down there," Tollie assured him.

"I still don't like it," Mouse grumbled. "Cats!"

"We won't land on the island, Mr Mouse," Maddie said, offering her own assurance. "They can't hurt you."

"There are so many of them, running all over the place. That poor sheep!"

"Now that you bring it, up, Mr Mouse, that is an interesting question about this island. Why don't the lions eat the sheep? They don't have to, they have enough grass to eat to live on. But they don't eat the sheep, which surely would be tastier for them. Why not?"

Mouse was not much mollified, but Tollie took an interest now. He wondered why, indeed, the lions did not simply eat the sheep.

Maddie read his expression. "There is a special condition that exists on this island. If a lion eats a sheep, he turns into a sheep, and then he could be eaten by another lion. The problem is why they cannot eat the sheep. Since the sheep is there and alive, why not just eat it? Perhaps the lion that ate it and turned into a sheep would be eaten as well, but perhaps not. How do they know whether or not they can eat that sheep and get away with it? There are a hundred lions and they are not eating that sheep."

"Yes," Tollie said, seeing the reasoning behind the question. "How do they know whether or not they would be safe if they ate the sheep? Since they are not eating the sheep they must have figured out it would not be safe to do so. But why?"

"Exactly," Maddie said.

Mouse still looked miserable, barely able to look at the island with all those big cats roaming over it. "I'm just glad they're not eating that poor little sheep! But why not?"

Tollie now recalled young Carl Friedrich Gauss and his solution of the addition problem. There might be a sequence that would explain why the lions did not eat the sheep. There were one hundred lions and one sheep... "What if there were one lion and one sheep?" he asked out loud.

Mouse groaned. "Then he would eat the sheep," he said plaintively. "He would turn into a sheep but that would not matter, because there would be no other lion that would eat him in turn."

Maddie was realizing something, too. "If there were two lions and one sheep..." she ventured, leaving the thought hanging.

"They would not eat the sheep," Tollie concluded for her. "If one of them ate the sheep and turned into a sheep, the other lion would eat him in his turn!"

Maddie smiled. "If there were three sheep, then a lion could eat the sheep and be safe, because you would be back at the case where there are two lions and one sheep and the sheep is safe!"

Mouse seemed heartened by this, but he was still too disturbed by the mere sight of the lions, the big cats below, to fret much about following the argument through.

Tollie took mercy. "Let's leave this area," he suggested to Maddie. "I think we have seen all we need to see and we have our answer to why the sheep is safe."

To Mouse's vast relief, they headed away from the island, and he relaxed as the distance from it grew, until he was comfortable again.

Gathering his wits again, he took on the completion of the reasoning.

"One lion, one sheep, the sheep is not safe. Two lions, one sheep, the sheep is safe, because if a lion ate it, he would be back at case one, one lion and one sheep, and the sheep (which he has now turned into) is not safe.

"If you extrapolate that you will find a pattern."

"Yes," Tollie agreed. "That is how Young Carl would have done it. "Extrapolation is extending the sequence out. One lion and one sheep, the sheep is not safe. Two lions and one sheep, the sheep is safe. Three lions and one sheep, the sheep is not safe. Four lions and one sheep, the sheep is safe because eating the sheep would take it back down to three lions and one sheep – the lion that ate the sheep and turned into a sheep would not be safe... Take it out to one hundred lions and one sheep."

"The sheep is safe," Mouse concluded. "It is the same case as two lions and one sheep. Given any even number of lions and one sheep, by extrapolation the sheep is safe."

"That is why," Maddie agreed. "If a lion ate the sheep and turned into a sheep, there would be 99 lions and one sheep, and he would not be safe. If there is an odd number of lions and one sheep, the sheep is not safe. Because one hundred is an even number, the sheep is safe."

Mouse smiled at this. He felt much sympathy for the poor sheep, alone among all those big cats. He was still grateful he was not the sheep. Even though he would be safe, he just would not be comfortable on that island full of cats!

### CHAPTER 14

WHICH DOCTOR?

The Island of Doctor Yes, Doctor No, and Doctor Maybe

"What is that island, the one in the distance that glitters so?" Tollie asked, having noticed something of interest on the horizon.

"That is the Island of Doctor Yes, Doctor No, and Doctor Maybe," Maddie told him. "They are brothers, triplets in fact."

Tollie thought it odd that they should encounter yet another set of triplets!

"They live together because they were born with certain conditions that would make it hard for them to be separated," Maddie went on. "First, they must speak in turn, the same doctor cannot speak twice in a row. Doctor Yes, Doctor No, and Doctor Maybe must each speak before Doctor Yes can speak again. In each case, each of them can say only one sentence.

"Then, they can say what they want except when they are asked a 'yes-or-no' question. In that case, Doctor Yes is only allowed to say 'Yes' or 'Maybe,' Doctor No is only allowed to say 'No' or 'Maybe,' and Doctor Maybe is only allowed to say 'Yes' or 'No.' Since they were born in the Village of Truth, which you visited just a while ago, they must still tell the truth insofar as they can, given these conditions."

"Interesting," Tollie said thoughtfully. "A very interesting situation. And, apart from when they are answering a yes-or-no question, they can say whatever they want, as long as it is true, since they are truth-tellers from the Village of Truth. But they must speak in turn, one sentence at a time. If they are to communicate at all with the outside world, they must stay together."

"That is the nub of it, and the rub of it," Maddie confirmed. "Now I am going to drop you off on their island. I'm afraid that I will have to leave you there. I have to take my whirlybird back home now, for routine maintenance. But you should have a good time with the doctors. They are entertaining fellows."

"Oh, I see," Tollie said. "Well, Maddie, thank you for the ride. I thoroughly enjoyed it!"

"It was a pleasure," she said. "And I will miss both of you, but I expect I will see you again another time."

"You will see me, I am sure," piped in Mouse. "I am always around."

"Of course you are," Maddie said. "I see you all the time and you are the absolute best!"

Mouse blushed, an odd thing considering he was a gray mouse.

Maddie landed her helicopter and took her leave. "You just have to walk up the hill, to the house of the three doctors. They will be gracious hosts and you will enjoy yourselves there."

"Thank you, Maddie," Tollie said, always polite. "I hope to see you again."

Mouse, still blushing, just smiled at her.

They watched as the helicopter whooshed into the air and headed off over the vast ocean, becoming just a wisp of motion in the air. Tollie glanced at his watch and saw that it was noon. Time was surely relative here. He knew it must still be nighttime in the "real" world.

Turning, they headed toward the path leading up the hill to the impressive mansion of the three doctors. Now that they looked at it more thoroughly, they noted how modern and spacious it was. The glass walls glinted in the sunlight. Facing them and the ocean was the broad, curved window that they guessed would have been for the living room, as it would have the best view from the hill. As they walked up the path and drew nearer, they could see that the curved glass windows were darkened, no doubt to shield the interior from the midday sun. Mouse and Tollie had started up the hill at noon, so the overhead sun would be quite bright viewed from the home of the doctors, especially that one wall, situated as it was to give the best view. Later on, they learned that the glass would lighten up, much like a pair of eyeglasses that would be shaded when the light was bright, but clear when it was darker.

Making their way up the path, they could see that a lot of attention had been paid to its design as well. A waterfall paralleled the path, cascading down in sections, divided by pools formed on the way down, each spilling over to the next waterfall and thence to the next pool. The pools were surrounded by landscaped small gardens with inviting resting areas, so one could pause on the way up to enjoy moments of tranquility. One could tell that the doctors were the kind of men who paid attention to the details that added together to make a world of their own, one created to induce a calm sense of wonder and enjoyment that made time seem to slow so that these simple elements were experienced in a leisurely pace.

About halfway up, Tollie and Mouse stopped at a particularly beautiful pool and rested on the bench placed there for that purpose. They listened to the water falling into the pool and watched it flow off at the other end. It was an idyllic interlude for them and they felt fully refreshed when they started again up the path.

Arriving at the house, they were greeted by the three doctors at the top of the path. They were tall, slim, elegant men, wearing crisp white cotton smocks, as befitted doctors, and white cotton slacks, white shoes, and no socks. Since they were outside, they also wore panama hats, perched jauntily atop their heads at an angle. Practical garb for the lush tropical isle they inhabited. They smiled pleasantly and gestured welcomingly to their home. Tollie thought they were quite a contrast to their encounters with the other triplets. One would take these men more seriously, but, still, they were very pleasant and well-mannered and one instinctively liked and trusted them.

"We have been expecting you!" said the first doctor, "Doctor 1."

"Yes," said Doctor 2, "and you are very welcome."

"We have been watching your travels in the World-Wide Web," said Doctor 3.

Tollie remembered what Maddie had said. They had to speak in turn, one sentence each.

"How so?" he asked. He wondered how they had been watching him and Mouse as they went from adventure to adventure.

It being Doctor 1's turn, he answered. "We monitor the World-Wide Web to make sure that the places you go to are safe for you."

"Yes," continued Doctor 2. "there are things like viruses, worms, Trojan Horses and bugs that can cause you a lot of trouble if you are not careful."

"We try to make it easier for you by watching out for them and preventing them from interfering with your travels," concluded Doctor 3.

Tollie looked at Mouse questioningly. "This is true, Mr Mouse?"

"I'm afraid so, Tollie. I can only take you where you want to go, I cannot protect you from what may be there when you arrive. Doctor Yes, Doctor No, and Doctor Maybe are computer programming experts. They devise programs that can protect you from other programs, developed by more wicked people than are you, I or the good doctors."

"It is perfect work for us," said Doctor 1, "given the speech rules we have to follow."

"This way we can work without having to be among a lot of people, and we are still doctors, so to speak, as our work is curing," Doctor 2 elaborated.

Doctor 3 summed it up. "It would be hard for us to work among a lot of people, and, of course, all three of us would have to work together, or we literally could never finish saying anything."

Doctor 1 changed the subject, as it was time to let their guests know the house rules. "You are very welcome here and we will do our best to show you a good time, but there is something you must first do to stay here."

"Yes," Doctor 2 took up the explanation, "anyone who comes here and is going to stay, must first identify us properly."

"That is," Doctor 3 said, "you must figure out which of us is which, Doctor Yes, Doctor No, and Doctor Maybe."

"And you cannot directly ask us to identify ourselves," Doctor One elaborated.

"You cannot ask 'Are you Doctor so-and-so?' directly, in other words," Doctor 3 explained.

At that moment their attention was diverted by a one-foot high purple elephant galumphing through the room. He was a cute little fellow and Tollie and Mouse were quite entertained by the sight of his lumbering elephant form, flapping ears and his trunk swaying to and fro. He went to a bucket in the corner and took his fill of water, expelling it mightily toward them, so that they felt a fine spray across their faces. He then trumpeted his satisfaction, a fine strong baritone emanating from such a small creature, though he was surely intended to be much larger!

"He is 'Li'l Jumbo,' our house mascot," Doctor 1 explained.

Li'l Jumbo galumphed back from whence he came, giving wide berth to Mouse. Tollie recalled that some elephants were indeed afraid of mice and laughed. Mouse smiled, as he was perfectly harmless and would not hurt a fly, or even a cat (though he did avoid them), much less a little elephant.

The three doctors smiled at their amusement upon seeing the little elephant, a rare sight. "Li'l Jumbo is quite a character and we treat him like a member of the family," Doctor 2 said.

Going back to the challenge presented them by the doctors, Tollie said to Mouse, "I am thinking again of the Apples and Oranges challenge. As with the three doctors, we have three choices, so if we identify two of the doctors, as with the Apples and Oranges boxes, we will automatically be able to identify the third."

"That's right," Mouse said. "We need only identify two doctors and then we will know which the third is. But what questions will we ask"

"Li'l Jumbo just gave me an idea about that," Tollie answered. "Here, have a look at this."

Tollie proceeded to draw a chart, in the form of a grid. Then another one. "These will outline some questions and their possible answers, which will identify our doctors."

Mouse took a curious look at Tollie's grids. His questions, listed in the grids, indeed concerned Li'l Jumbo, and could in no way be taken to be referring to the doctors themselves, directly or indirectly. So they were within the rules.

### Grid 1

How Doctor Yes, Doctor No and Doctor Maybe would have to answer:

(Remembering that the doctors must answer as truthfully as possible, given their limitations.)

### Grid 2

How the answers of Doctor 1, Doctor 2 and Doctor 3 would identify them:

Tollie explained.

"YYN is a possible sequence of answers for the first three questions. If so, we know Doctor 3 is N or M, as he answered "no." So he must be N, since Doctor 1 and Doctor 2 could only be Y or M and, therefore, M is not available for Doctor 3. We can see this by looking at Grid 2. We know Doctor 3 is N, but we do not know whether Doctor 1 and Doctor 2 are Doctor Y or Doctor M, as this is still indeterminate, they could be either. Therefore, we require a fourth question.

"Another way of looking at it is that one of the doctors must answer twice if we are to get a comparison that would identify that doctor. One answer from each doctor will not give enough information to identify them or any one of them. Once a doctor gives two answers, however, his identity is known, as can be seen by examining Grid 1 above.

"Since they must tell the truth, insofar as the rules allow them to, the optimum strategy is to vary the questions between those that assume a true premise (is it an elephant?) and those that assume a false premise (is it a dog?). This will force them to answer differently according to the rules they each must follow, allowing us to deduce which Doctor is which.

By looking at Grid 1, we can see that a "yes-maybe" pair of answers to questions 1 and 4 means it is Doctor Yes. A "maybe-no" pair means it is Doctor No. A "yes-no" pair means it is Doctor Maybe.

There are three possibilities. Once we know one doctor's identity, we can examine the grid to determine who the other two are.

If Doctor Yes, the doctor who answered "maybe" to question 2 will be Doctor No.

If Doctor No, the doctor who answered "maybe" to question 3 is Doctor Yes.

If Doctor Maybe, the doctor who answered "yes" to question 2 is Doctor Yes.

Once we know the identities of two of them, just as in the case of the boxes of Apples and/or Oranges, we will know the identity of the third doctor.

The three doctors nodded approvingly. "Quite logical," Doctor 3 said, as Doctor 2 had been the last speaker. The good doctors had good memories.

"I especially like the way you made use of the grids to solve the problem." Doctor 1 said.

"Carl Friedrich Gauss would have approved," said Doctor 2.

"You know about Carl?" Tollie asked in wonderment.

"Oh, yes," affirmed Doctor 3, "we had to approve him for the World-Wide Web you explored, to make sure he was the real Carl Friedrich Gauss."

"Of course," Tollie said. "I see."

Doctor 1 extended his hand. "I am Doctor Yes."

"And I am Doctor No," said Doctor 2.

It was Doctor 3's turn. "You're method was excellent, so we might as well let you know without making you go through all the questions, and I am Doctor Maybe, of course."

"Didn't you just sneak in two sentences in the guise of one?" Tollie asked mischievously.

"Maybe," said Doctor Yes, equally mischievously. Of course, if he had passed his turn, which would acknowledge that Doctor Maybe had taken two turns, Doctor No would also have said "Maybe," Tollie realized. Hoist by his own petard, he grinned.

The doctors waved them into their house and they followed in anticipation. They knew by now that it would be an interesting experience.

It was a big shambling house of many rooms, each with its own character. They entered the modern living room with the expansive curved view, but, beyond, they could see bits and pieces of other rooms of a quite different character. There was a library visible, for example, containing full walls of bookshelves filled with new and old books of all sorts, with sliding ladders arranged to allow access to the higher volumes.

From the expansive window, they could see the ocean, much of the island, and, very clearly, the path they had ascended to get here, including even the pond where they had stopped to rest. They could see, too, that this path was the only way to get to the doctors' house. So they would always know in advance that someone was coming.

Tollie's eyes returned to the library. All those books! They fascinated him, for some were visibly old books, and he loved old books. They had knowledge from ages past that had either developed into our modern knowledge or had been mostly forgotten. But they were still of interest, since they gave an idea of how we as people thought of the world through time.

Seeing his interest, Doctor No said, "Please, you are very welcome to come and see our library."

Doctor Maybe: "Yes, of course, we must show it to you."

Doctor Yes: "Come, come!"

### CHAPTER 15

MR QUIRM

Many a Quaint and Curious Volume of Forgotten Lore

Entering the library, the three doctors, Mouse, and Tollie were greeted by a long smiling creature, small enough to fit in your hand, but long enough to wrap itself around your thumb. He was perched on a shelf directly in front of their noses.

Doctor No: "Mr Quirm, allow me to introduce our new friends, Tollie and Mr Mouse."

"Hello, Mr Quirm," Tollie greeted the creature, politely as was his way.

"Mr Quirm," Mouse acknowledged.

Doctor Maybe: "Mr Quirm is our librarian."

"That I am," Mr Quirm agreed. "My name is Booker Quirm, but they call me Book for short, Book Quirm. As you can see, my name is similar to my occupation. I am a bookworm. I could simply devour volumes and still want more books. But do not worry, I only eat those parts of books that have no print, mainly the margins." He giggled merrily at the thought of all those delicious musty pages. "What could be a better career for me than being a librarian?"

Doctor Yes: "Mr Quirm keeps all our books in order so we can find what we want among them."

Doctor No: "In exchange, we let him eat away to his heart's content."

Doctor Maybe: "As long as he keeps his promise and eats only those parts of the page that are without print." He gave Mr Quirm an admonishing glance.

"I am about to have lunch, as a matter of fact," Mr Quirm announced. "So, you will see that I keep my promise. I only eat through the blank margins of the pages. They are more delicious anyway! None of that ink. Ink is yukky."

He performed a comic side-winding slide toward a section of the bookshelf on which he had been perched, approaching an imposing looking ten-volume set.

"That is lunch, I presume," Tollie queried.

"Oh, yes, and a feast it will be!"

Mouse looked at the little fellow, wondering why he could not just eat cheese, like any sensible creature.

"Now, Tollie," Mr Quirm continued. "I have been watching, along with the good doctors, your progress through our World-Wide Web and the many challenges you have faced and solved along the way. I now have a challenge of my own for you."

"Oh?" Tollie felt a sense of wonderment that even this small but friendly enough worm would present him with a puzzle, if puzzle was what it was to be.

"I am going to eat my way through these ten volumes for lunch. It is a ten-course meal, you might say."

Tollie noted a certain pomposity in the little worm, but tolerantly accepted it. Mr Quirm, was, after all, proud of his well-developed gastronomical tastes.

"Your challenge is to tell me how far I shall travel as I eat my way through the ten volumes, from page one of volume one to the last page of volume ten!" Mr Quirm announced all too portentously for his size. "Each volume is two inches thick, with the covers being one eighth of an inch thick each. You might say that the pages of each volume make up one and three fourths inches, while the covers take up one fourth of an inch. And if you said that, you would be right. Each volume would still be two inches thick in toto." He swelled his chest in pride, having showed off his skills in the dead language of Latin.

Latin was a subject Tollie had not found endearing. It was more something he had been enduring. Nevertheless, he put up with Mr Quirm's quirky verbal mannerisms. His interest was piqued by the puzzle presented.

"Wouldn't you be eating through 20 inches minus two covers, one eighth of an inch each or a quarter of an inch total, so you are traveling nineteen and three fourths of an inch?"

Mouse nudged him. "Think it through a little more, Tollie. Where is page one of the first volume and where is the last page of the last, the tenth volume? Think of the books as they are on the shelf and then as they are when you pull them out to read them!"

"Of course!" Tollie said, realizing it was not as simple as he had thought. "The first page of volume one is on the _right_ side of the book and the last page of volume ten is on the _left_ side... This is looking at them as they are on the bookshelf versus how they are when you pull them out and read them."

Mouse smiled helpfully. Mr Quirm literally squirmed, realizing the jig was up and Tollie _would_ solve his riddle. The good doctors smiled genially.

"Mr Mouse is very good to you, Tollie," observed Doctor Yes.

Doctor No: "You do realize he can only give you hints?"

Doctor Maybe: "But they help you along your way, even though in the end you are left to figure it all out yourself."

"Oh, I appreciate Mr Mouse very much indeed," Tollie allowed. "He is my guide and friend here."

Mr Quirm wriggled himself into a more comfortable position. He was smug enough, though, as he knew that in the end he would have his book feast.

Tollie proceeded. He had the measurement of the distance Mr Quirm would travel in eating his way through the set of books. "From page one of volume one all the way to the last page of volume ten, Mr Quirm will travel, as he eats his way along, a distance of sixteen and one fourth inches." He stated it baldly, as a fact. Mr Quirm looked a bit put out.

This is because page one of volume one is on the right side. He will eat his way through one eighth of an inch to get to volume two. He will eat his way through volumes two through nine, a total of sixteen inches. He will then eat his way through one eighth of an inch of volume ten to get to the last page of that volume. This is a total of sixteen inches for volumes two through nine. He eats that plus one eighth of an inch, the front cover of volume one, and one eighth of an inch more, the back cover of volume ten."

Mr Quirm set out on his meal. His riddle had been solved and he was hungry. And, yes, it would be sixteen and one fourth inches of savory, musty pages and covers that he would be eating. He set at his lunch with gusto, and they watched as his tail disappeared into the front page of volume one. They could hear him now as he started munching his paper trail.

Now Doctor Yes gestured toward the game tables. "We have all kinds of games here," he said.

Doctor No: "This one here is our chess table, the board is in the middle, and you can see we even have timers, so we can play official chess."

Doctor Maybe (gleefully): "But we have a new problem for you!"

Tollie, Mouse, and the good doctors moved over to the board.

### CHAPTER 16

CHESSBOARD AND DOMINOES

"Is it a chess problem?" Tollie asked. "I like them but I have trouble solving them."

Doctor Yes: "It is not a chess problem, but let us call it a chessboard problem."

Doctor No: "And a domino problem!"

Doctor Maybe: "Yes, a chessboard and dominoes problem."

Doctor Maybe gestured around the room now.

Tollie looked around and realized that the game room adjoined not only the library but also the living room. The great curved window swept through here also, and he could see the wonderful view the doctors had of the island, the ocean, and the path they had ascended to get here. It was an architectural accomplishment and somehow Tollie knew the doctors had designed the entire property themselves, the gardens and paths as well, as their home and personal workspace. Since they patrolled the World-Wide Web they needed a wide vista, so they could see the many things that were there to be seen. His attention focused on the chessboard on the table.

It was inlaid into the table, which was a normal game table, on which any game could be played. He noted the green felt top, which could cover the table if you wanted to play cards, for example. Or dominoes. He noted also that the doctors had many games stored in the under-structure of the table. They were clever fellows indeed. Chess, checkers, go, dominoes, backgammon, Scrabble, decks of cards, all were there. At present, the chessboard was blank. The chess pieces had not been set up.

Doctor Yes pulled out a decorative wood-inlaid box and slid its cover open, revealing a set of dominoes.

"Note that each domino covers exactly two squares of the chessboard," he observed to Tollie and Mouse. To illustrate, he covered two squares of the board with one domino.

Doctor No: "You see, a chessboard has 64 squares, 8 X 8, and we can cover it completely with the dominoes, using 32 of them, since each one covers exactly two squares."

"But," Doctor Maybe took up the presentation of their puzzle, "suppose we remove two squares from the chessboard, specifically the two diagonally opposite squares!" He said this with considerable glee, much as Mr Booker Quirm had presented his puzzle.

Tollie marveled that they spent their time with such entertainments as a matter of course. He was glad to live in the real world, he thought, even though he greatly enjoyed visiting the virtual World-Wide Web. "All right, he said, suppose we do remove those two squares?"

"Well then," Doctor Yes finally posed the challenge, "could we still cover the chessboard completely with the dominoes, using 31 now, since two squares have been removed?"

Tollie squinted at the board. He did not immediately understand the problem. "Why not?" he asked. "Each domino covers two squares and we have removed two squares, so we still have an even number of squares. 31 dominoes should do it."

"Well, let us see you do it," Doctor No challenged.

Tollie began covering the board with the dominoes, two squares at a time. But, try as he might, he found that no matter how he did it, the last domino was left without two squares to place it on. At first he thought it was a matter of his not arranging them properly. But after repeated attempts, he began to suspect there was some reason it could not be done.

"We have removed two squares," Doctor Maybe offered in explanation, "so why has it now become so difficult to cover the board with the dominoes?"

Tollie looked at the board again, assessing the situation. Two corners had been removed, that was all. Otherwise, it was the same. Why wouldn't 31 dominoes cover it?

"The diagonally opposite corners are the same color," Mouse offered, giving Tollie a hint, as was his job.

Doctor Yes: "And we can see that any domino placed on the chessboard will cover exactly two squares."

Doctor No: "And the squares on a chessboard alternate between black and white."

Doctor Maybe: "Soooo...each domino must cover one black and one white square."

"I see," Tollie said, though he was only beginning to see. "By removing diagonally opposite squares, we are removing two squares of the same color, say two white squares for example, leaving 30 white squares and the original 32 black squares and 62 squares in all..."

He left the problem hanging for a while, mulling it over. Booker Quirm, finished with his lunch, squirmed into the room oozing his gustatory satisfaction, glanced at the chessboard and gave Tollie a snidish grin. Mr Quirm had a small bit of malice in his heart, Tollie thought.

He sighed and returned to his contemplation of the chessboard. The "Aha" reaction dawned upon him. "By removing two white squares you are leaving two black squares unpaired, they don't have corresponding white squares to form the rectangle that would be covered by one domino. On a chessboard, remember, the domino can only be placed so that it covers one white and one black square. So there will be not one, but two dominoes that cannot be placed on the board, because they will not have corresponding pairs of squares of alternating color to be placed upon."

Booker Quirm slithered off, disappointed that again Tollie had handily solved the new puzzle. His mind churned over until he had yet a new challenge for Tollie. The same snidish grin stole over his round face. This one would stump him.

Doctor Yes: "Very good, Tollie, well done!"

Doctor No: "Indeed, by removing two squares of the same color you have made it so the board can no longer be fully covered by the dominoes."

Doctor Maybe: "We have many such puzzles here, it is our game room."

Mouse winked at Tollie and Tollie winked back. The others had not noticed Mouse's key contribution to the solution, but Tollie had. The estimable Mr Mouse was certainly well-traveled and there was not much that he had not encountered while serving as first-line guide in the World-Wide Web.

Booker Quirm now sidled over. "We like to play bridge," he announced portentously.

"I never learned the game," Tollie confessed.

"The doctors and I play often," Mr Quirm said, smiling at Tollie and Mouse. The doctors, all three, nodded the affirmative.

"How do you play?" Tollie asked. "I would think it would be difficult for you to hold the cards."

Booker Quirm now unveiled one of his surprises. From his sides, he revealed two tiny hands. "They are enough to hold the cards," he assured them. "And now, I have another little puzzle for you..."

### CHAPTER 17

NO BIG DEAL

At this moment, in gamboled Li'l Jumbo, trumpeting loudly through his little trunk. The doctors regarded him kindly as he stood in front of them, his baleful demeanor demanding attention. "He wants his water," Doctor Yes explained.

Doctor No went to the corner of the room and turned a spigot. Tollie and Mouse now saw that Li'l Jumbo had his own water fountain, which let out into a small pool that he could dip his trunk in and draw out as much water as he wanted. They watched as the tiny elephant had his fill. Li'l Jumbo used the bucket for his water only when the doctors had not turned on the fountain, Tollie noted to himself. He thought it smart of Li'l Jumbo to have a backup source of water. But, naturally, Li'l Jumbo had all the intelligence of a full-sized elephant.

If Tollie had any questions about how a worm could have hands even though that was a biological impossibility on the face of it, given that Mr Quirm had no feet, he now realized that it was perfectly possible here, if not in his own world. There were one-foot high elephants here!

The worthy gentleworm, Mr Quirm, cleared his throat for attention and Tollie and Mr Mouse turned from watching Li'l Jumbo to paying attention to the self-important worm. After all, they were guests in the home of the doctors and if the doctors had Mr Quirm in their employ, it was their duty to be courteous to him.

"We were playing bridge here at this very table," Quirm proceeded, satisfied that he had their undivided attention. "I was the dealer." He waved his small arms and wiggled his yet smaller fingers, showing that he could, in fact, play cards.

"I dealt out about half of the deck when the telephone rang. I left the table to answer."

Tollie blinked. He could not for the life of him picture Mr Quirm handling a telephone. But it must have been so. The good doctors were not fazed at all at the idea of a worm answering the phone. Quite the opposite, Tollie got the impression that it was natural to them that this was one of Quirm's duties here. And why should not the resident librarian answer telephone inquiries?

"When I returned to complete the deal, I found that I had only dealt half the cards, _and_ I could not remember to whom I dealt the last card!"

Why this was so monumental a discovery escaped Tollie's understanding, but he remained, as was his way, courteous, and waited for Quirm to finish.

"I could have reshuffled, then deal the cards out as usual. But, I realized that there was a way I could continue with the deck as dealt, and deal the rest of the cards out so that everyone would get the same cards they would have if I had not been interrupted by the telephone call. Now... How did I do this?"

He puffed up his chest, clearly proud of his accomplishment in logical reasoning and his timesaving solution. He was not a worm to waste time. He was an early worm.

Tollie scrunched up his face a little, something Mouse had not seen for a while, but the thing was that he did not quite realize what the challenge here was. "He could not remember to whom he had dealt the last card," Mouse said, hoping to help Tollie see the question posed by the worm. "Then he was able to continue dealing, without having to start over, but guaranteeing that each player would receive the same hand as they would have if the deal had not been interrupted."

"So he came back to the table, not knowing who got the last card, then dealt out the rest of the cards so that everyone got the same cards they would have gotten..." Tollie said, considered this. "Bridge is a game of four, there are 52 cards in a deck. 52 is divisible by four. In other words, in a regular deal, the dealer would get the last card."

Mouse danced a bit, humming under his breath, and Tollie took this to mean he was on the right track. Indeed, he now saw the solution. He also saw Quirm squirm, as sure sign that even he could tell that Tollie would ultimately solve his riddle.

"The dealer would get the last card, which is the bottom card! Of course! He deals out a few cards, goes and answers the phone and comes back. Then he deals from the bottom of the deck in reverse, dealing himself what would have been the last card first. The second to last card would go to the second to last player, as it would have if the deal had gone through in a forward direction as usual. By extrapolation, the third to last card would go to the third to last player and so on through the deck until the last card went to the player who would have received it anyway if the deal had gone through forward from the top instead of backward from the bottom."

Quirm slinked away a distance, crestfallen. Again, Tollie had solved his challenge.

But Tollie smiled at him. "Mr Quirm, it was a wonderful puzzle. I was completely stumped by it at first. I didn't even see it as a problem that could have a solution. You should not be disappointed.

Mollified, Quirm smiled a bit, almost shyly, then grinned broadly. "You did great, Tollie, with both puzzles. I was just being hard with you, maybe because you were a stranger here. But now I see that you belong and are welcome here. You are one of us."

"Well, thank you very much, Mr Quirm. I'm happy to be here. And I did enjoy your puzzles very much. I will remember them and they will bring joy to others as well."

Quirm blushed. This was something none of them had ever seen, a blushing worm! It was more curious than seeing Mouse blush. Returning from his fountain, having drank his fill, Li'l Jumbo snorted. Quirm was almost as purple as he was! Worms look purple when they blush, Tollie thought, making a mental note to remember this, too. It could come in handy in a trivia quiz. Li'l Jumbo came as close to laughing out loud as an elephant, even a tiny one, could.

### CHAPTER 18

PICK A NUMBER, ANY NUMBER

Doctor Yes, Doctor No, and Doctor Maybe joined in the laughter.

Doctor Yes: "We have one for you."

Doctor No: "All you have to solve is how it is done."

Doctor Maybe: "Yes, No is right, just figure out how we do it."

Doctor Yes: "Pick a number, any number."

Doctor No: "Both of you pick a number."

Doctor Maybe: "That way you will see that it truly works for any number."

Tollie and Mouse thought up numbers, each on their own. "I have a number in mind," Tollie announced.

"And so do I," Mouse added.

Doctor Yes: "Multiply your number by seven."

Tollie and Mouse duly multiplied their numbers. It took Mouse a little longer, so you can guess he had originally picked a larger number.

Doctor No: "Now add 11 to it."

That was easy enough.

Doctor No: "Remember the number you now have, file it away so you can recall it later."

"Done," said Tollie, and Mouse nodded as well.

"Doctor Yes: "Now, add the digits in that number together."

Doctor No: "Yes, for example, if your number is 31, you get four when you add the digits together."

Doctor Yes: "Subtract the number you now have from the number No told you to remember."

"Yes?" Tollie had done the math, but he wondered where this was going.

"Well, well..." said Mouse.

Doctor No: "Okay, now add 23 to the number you now have."

They did so, taking a little longer than before. They were working with larger numbers now.

Doctor Maybe: "Add the digits in that number together."

"Yes," Tollie said and Mouse said, "Ditto."

Doctor Yes: "Do you have a one-digit number."

"No," said Tollie. "Yes," said Mouse.

Doctor No: "Tollie, add the digits in your number together again."

Doctor Maybe: "And you, Mr Mouse, can just work with the number you have, since it is a one-digit number."

Tollie added the digits together again. "I have a one-digit number now," he offered.

Doctor Yes: "Excellent!"

Doctor No said with a flourish, "The number you now have is five!"

Tollie and Mouse looked at each other for confirmation. Of course, it was so, they both had five!

Doctor Maybe: "Do you think No is a mind reader?"

"No," Tollie said. "I think it is a number trick, but I don't see how it is done."

The doctors laughed, happy that they had pulled off their trick to the mystification of Tollie and Mouse. But they did not realize they had underestimated Mouse...

"I have noticed," said that worthy, "that when you take any two-digit number, add its digits together and then subtract the result from the number, you end up with a number that is a multiple of nine."

Tollie did some quick mental math and realized that this was, in fact, true. He even went beyond this and saw that it held for three-digit numbers as well, and he suspected it would hold true for any multiple-digit number.

"A two-digit number has two columns, the tens and the ones," he ventured forth in his effort to explain the number trick. "If it is 34, for example, and you subtract 4, you are taking the ones away, leaving it at 30. If you now take away 3, representing 3 tens, you will get 27, a multiple of nine. 34 minus the sum of its digits, 7, will give you 27, a multiple of nine."

Doctor Yes: "That is correct, Tollie."

Doctor No: "Another way of representing that is that 30 is three times ten, and three times nine would be three less than three times ten, since there is a difference of one between nine and ten."

Doctor Maybe: "Let me write that down so we can see it."

He wrote:

3 x 10 = (3 x 9) + 3, or

3 x 10 = 3 x (9 + 1) = (3 x 9) + (3 x 1) = (3 x 9) +3

"So," Mouse took up the explanation, thinking at least he could say more than one sentence at a time, and Tollie might appreciate that at this point, having listened at length to the three doctors alternating their sentences, "a number times nine will become that same number times ten, when you add that number again. Four times nine plus four equals four times ten. Four nines plus four equals four tens. Seven times nine plus seven will become seven times ten, and so on, again by that method we have learned is called 'extrapolation.' Then, you add to that again the number of ones that exceed the multiple of ten."

"Yes, I understand," Tollie said. "It sounds harder than it is, but if you write down some numbers, it becomes clear and obvious. Take any two-digit number, let's say 67 as an example." Tollie wrote it down, to illustrate.

67 = (6 x 10) + 7

"Adding the digits together and subtracting, gives you this."

6 + 7 = 13 and 67 - 13 = 54

"54 is a multiple of nine, as it has to be, given what we have already observed. Six tens plus seven ones gives you the difference between 67 and 54, the multiple of nine. This is true of any two-digit number."

"That's perfect," Mouse said. "It will also hold true for higher numbers than two-digit numbers. If it were a three-digit number, you would be adding the number of hundreds, plus the number of tens, plus the number of ones, and you can extrapolate this to numbers of many digits – the number of thousands, plus the number of hundreds, plus the number of tens, plus the number of ones, all the way to the very largest numbers."

Tollie considered this. "To complete the doctors' number trick, you just reverse the process, once you have a multiple of nine. That is, a multiple of nine plus five will give you a number whose digits add up to five. Take 27 as an example. 27 plus 5 equals 32, and adding the 3 and the two gives you 5.

"But the doctors added another wrinkle. They added 23 instead of 5. But the digits of 23 add up themselves to 5, so it is the same as adding 5."

"That's right," said Mouse. "And that is because 23 is itself a multiple of 9, that is 18 in this case, plus 5. So all you are doing is removing the multiple of nine, giving still a multiple of nine, after the 5 is accounted for."

"Yes," Tollie said, seeing the whole thing now. "Take any multiple of 9. Let's say 36 – add 5 and you get 41 and 4 plus 1 add up to 5. But, add 23 (which itself adds up to 5) to 36 and you get 59, wherein 5 plus 9 adds up to 14, and, since you do not have a one-digit number yet, you add the digits together again, and the 1 plus the 4 again add up to 5. No matter what number you chose to start with, the doctors can get you to 5, or, indeed, to any number they wish."

Mouse hummed his little tune. The trick was revealed.

The doctors smiled. They knew Tollie could figure it out eventually, but they had not realized that Tollie and Mouse together could do it that much faster.

Now Li'l Jumbo lumbered into the room, took up station in the middle, and regarded them sideways with a friendly elephant eye. They all looked at him and he nodded his head up and down to show that he knew he had their attention.

### CHAPTER 19

THE MONTE HALL PARADOX

"I have a teasing little problem for you," announced Li'l Jumbo.

Tollie and Mouse jumped. They had thought that Li'l Jumbo could not speak. Neither could have explained why they thought that, perhaps it was just because Li'l Jumbo had not spoken before. But they had encountered all kinds of animals that had spoken to them, so there was really no reason the little elephant could not also speak.

Evidently, Li'l Jumbo was over his fear of mice now, as he addressed Mouse along with everyone else perfectly naturally.

"I like to watch old TV," Li'l Jumbo told them. "Even old game shows. There was one called 'Let's Make a Deal," which was hosted by Monte Hall. I watch that one a lot and I have heard of a paradox related to it. You might know the show."

Mouse nodded but Tollie had to admit he had never seen it. It was before his time.

"It is a show," Li'l Jumbo explained, "in which contestants were presented with deals by the host, Monte Hall. These deals usually involved a choice between something they knew, and knew what its value was, against something in a box or behind a door, that could be nothing, something of lesser value, or something of greater value. They could stay with what they had, but they had a chance to get something even better."

"I see," Tollie said. "They could lose it all or they could improve on what they had."

"Yes," said Li'l Jumbo. "Now, the paradox I heard is about a contestant who has a toaster oven already, but is offered a prize of considerably greater value that is behind one of three doors, perhaps an automobile. He has a one-in-three chance of getting a car, so he goes for it and picks door number 1."

"I would probably do the same. A toaster oven is nice, but having a one-in-three chance of getting a car is tempting."

"Yes. So, he has picked door number 1, but... Monte Hall now opens door number 3 and reveals that there is nothing behind it. He asks the contestant if he would like to change his choice from door number 1 or stick with door number 1. The question is: Should the contestant stick with door number 1 or switch his choice to door number 2. He now knows there is nothing behind door number 3."

"It doesn't matter," Tollie said. "He now has two doors each with a fifty-fifty chance of having a car behind it. He could switch or stay with door number 1."

Mouse had a small but gracious coughing fit, which Tollie took to be a hint that he had answered too quickly. But was there more to this than he thought? Tollie thought it over.

It seemed that his logic was right. At first, with a choice between three doors, the odds of a car being behind door number 1 were one-in-three. With door number 3 revealed to have nothing behind it, the odds were now one-in-two, or fifty-fifty.

So it did not matter which door he picked, each would have a one-in-two chance of having a car behind it. He glanced at Mouse, wondering if Mouse had gone astray somehow.

Mouse admonished him, "Think of the first situation, where all three doors are closed. The odds of a car being behind _either_ door number 2 or door number 3 would be two-in-three while the odds of it being behind door number 1 would be one-in-three."

Tollie thought this over. It was right, of course, but what did it have to do with the problem at hand? Surely the situation changed when Monte Hall opened door number 3 and revealed that there was nothing behind it!

Or... did it?

If it didn't, why not? Tollie was thinking now. Applying the principles of reasoning to the problem. It was something he had learned to do in the course of his travels in the World-Wide Web and he realized as he was doing it how wonderful it really was! He now understood better the meaning of Young Carl's advice. The words came back to him:

"I always believe there are answers, it is just a matter of finding them. But believing in the reality of there being an answer is always the first step. Even if there is not an answer, you have to show why not. And you can think your way through it, by patiently applying logic and reason to the question at hand."

This was indeed what Tollie was now doing and it seemed second nature to him. It had not been that way before he entered the World-Wide Web, met Mouse and embarked upon their adventures. The realization made him happy. It had turned out to be a wonderful birthday present.

Too, he realized something else. Just because Monty Hall had revealed that there was nothing behind door number 3 did not change the odds, the probabilities. There was still a two-in-three chance that a car would be behind door number 2 or door number 3, the only thing that had changed was that now he knew there was nothing behind door number 3. But that knowledge did not change the situation, which remained the same.

"Door number 1 has a one-in-three chance of having a car behind it," he announced, a little proudly. "Door number 2 and door number 3 have a two-in-three chance of having a care behind them. Now that we know door number 3 does not have a car behind it, we also know that door number 2 now has a two-in-three chance of having a car behind it! Door number 1 still has only a one-in-three chance so we should switch our choice now to door number 2!"

Li'l Jumbo did a little dance now!

"One-foot, one-foot high am I,

One-foot high elephant,

One-foot high..."

He, too, had his song! Tollie wondered yet again about this, the song of the World-Wide Web... He found it natural, as it should be. The song of life, the celestial music of the spheres... Here in this brave new world.

"Oh, Tollie, you have done well!" said Li'l Jumbo, winding down his dance. "If you doubt it, think of this. You have ten doors rather than three and chose door number 1. Then eight of the other doors are opened, all with nothing behind them. You are left with door number 1 which you chose originally, and door number 2. Door number 2 still has a nine-in-ten chance of having a car behind it, but your choice, door number 1, still has only a one-in-ten chance!"

"Yes, that is true," said Tollie. "You have extrapolated it right."

Li'l Jumbo took up his dance and trumpeted mightily his satisfaction, producing a sound worthy of a full-sized elephant. He was clearly proud of himself as well and gave a bow, going down on one knee to do so.

Mouse smiled at the little elephant. Tollie thought it interesting in that Mouse was four-foot high and Li'l Jumbo only one-foot high. What a world it was... Yet, Mouse and Li'l Jumbo could get along here.

The three doctors now stepped forward.

Doctor Yes: "Come, come!"

Doctor No: "Yes, there is yet more here for you to see."

Doctor Maybe: "Come, come!"

Tollie and Mouse took their leave of Li'l Jumbo, who trotted off happily to his personal water fountain. Tollie and Mouse left with the good doctors, as they did not want to wait too long. Li'l Jumbo might spray them again!

### CHAPTER 20

REFLECTIONS ON THE CIGAR GAME

They entered the game room, or simply moved over to it, as it was really a part of the living room and library, something Tollie had not noticed before. It was a large L-space, with the library occupying the short arm of the L and the living room and game room taking up the long arm. The space was deceptive as each area could have been a large room by itself. Adding to this the high ceilings, one could be forgiven for not at once realizing it was one continuous space. The curved window took up the outer edge of the long arm of the L, discombobulating that honorable letter's form by only a small warp.

It was a magnificent game room, awesome in content as well as dimension, having ample room for billiard and pool tables, a ping-pong table, air hockey table, foosball, a card table and a chess table, which were the same as Tollie had seen before, though he had not realized that he had wandered outside the library then and into part of the game room. The high bookshelves of the library had obscured the game area. Also, the game room then merged right into the living room, as he could see more fully now, from his vantage point at the corner of the L. This entire L space was geared to learning and fun, the library, the game area and the living area all of one piece and laid out for convenience, yet seeming separate somehow. You had only to walk from one area to the next yet it truly did feel like you were entering different rooms.

He looked around at the walls, which were covered with illustrations of puzzles and enigmas of various sorts, geometrical and logic puzzles, optical illusions, art trickery, mazes, and metal and wood interlocking puzzles which could be disassembled and then reassembled if you could. There was a larger than usual Rubik's Cube constructed of different shades of natural wood that made a wonderful decorative piece, yet could still be manipulated into its millions of possible permutations.

Tollie could see now that an arch led from the living area into an ample dining room, capable of seating twelve comfortably and yet still a cozy space. "Eat, drink, and be merry," Tollie thought. And read and play games and dance as well, there being room for all manner of activity.

Having already seen the chess table and card table and realizing their situation between the library and the game area, Tollie now looked beyond the billiard and pool tables and saw a man sitting in the living area, on a couch in front of a rectangular coffee table. He was bulky and wore a pin-striped three-piece suit that contributed to the impression that he might have been a banker, but it was the gold pocket-watch with gold chain curving elegantly from his vest that completed the characterization. Cigars were scattered all around him, but he was not smoking, though it would not have seemed out of character.

He stood and walked briskly toward them, moving gracefully given his weight, and extended his hand.

"Hello, PJ Nagrom here, and who might you be?"

"I'm Tollie and this is Mr Mouse," Tollie said, extending his hand in return. "We are travelers and guests here, guests of the good doctors."

"Well, so am I," said PJ. "The good doctors are kind enough to be my hosts. And I, in return, am indeed their banker. You can bank on me!"

"They are our hosts as well and I take it you handle their financial affairs with great discretion," Mouse said, extending his "hand."

"Indeed, I am good at it." Ceremonies being observed and his role here clarified, with a certain pride of his own, PJ waved them over to the couch and the coffee table.

"I was just considering a new game," he began. "This excellent setting inspired me. I thought I would invent a new game. Would you like to play?"

"We would like to," said Mouse. "But I must let you know, my good friend here, Tollie, has become an accomplished player."

Now it was Tollie's turn to blush. He let it go, as there really was nothing for him to say. He had met a few challenges by now and was curious to see what PJ would propose.

"I have been thinking about a new game, a cigar game, on this very coffee table." PJ gestured expansively, encompassing the coffee table, the couch, and all his cigars strewn about. But I have a problem. It seems to me that it is possible that one player will win while the other must surely lose. Either the one who moves first or the one who moves second. I am not sure."

"What do you mean?" Tollie asked, his curiosity piqued. "Please tell us the rules of your new game." He now saw a glimmer of a reason for all those cigars surrounding the heavy banker.

"The players take turns laying cigars on the coffee table. You notice that the cigars are standard, they have a rounded end and a flat end. So the players can place the cigar by lying it down or they can stand it up on the flat end. The object of the game is to be the last player to lay a cigar on the table before it runs out of space. Is there a move that the first player can make or the second player can make that will guarantee victory for that player?"

"Wouldn't they have to play it out to see?" Tollie asked. "It is a normal-sized coffee table, a lot of cigars would fit on it. How could you tell the outcome in advance?"

"Perhaps," Mouse suggested, "there is a strategy that one or the other player can use that will win the game. An algorithm, if you will."

"There is also the fact that there are two ways of placing the cigars, one flat and one standing on end, balanced on the flat end of the cigar," Tollie reflected. "So there are a lot of combinations of these up-and-down and flat placements. You are asking for a strategy, a method of placement, an algorithm as you call it, that, if followed throughout the game, will guarantee victory to one or the other player."

"That's right, Tollie," said PJ Nagrom. "I have a feeling there is such a strategy, but I cannot figure it out." He paced up and down, moving gracefully given his bulk, deep in thought.

"There is only one position on the table that cannot be duplicated on the other side of the table," observed Mouse quietly. "That would be the center. Any other position would have a corresponding opposite position."

Tollie and PJ thought this over, its relevance becoming slowly clear. It was to be a hint to Tollie, but it really revealed the whole thing, something unusual for Mouse, who usually gave small hints. Tollie was working out a strategy based on Mouse's observation.

As PJ paced back and forth, Tollie gave voice to his thoughts on the solution. "I believe one player could win if he made the right first move. That move would be to place the cigar on end, standing up, in the center of the table, since the center is the only position that cannot be duplicated on the opposite side of the table. The second player would place his cigar anywhere on the table. Now the first player has only to duplicate this move on the opposite side of the table, that is, make the reverse move."

"Yes, yes," said PJ Nagrom excitedly. "The first player can always play a cigar on the table, since the second player's move will always have its equal and opposite position. When the second player plays the last move on one side of the table, there will be a corresponding last move on the opposite side, so player one will always have a position to place his cigar and will be the last one to have a space to place it on. He wins the game."

"But," Tollie added, "player one will also have to reverse the cigar itself. If it is laid down along its length with the rounded end pointed in one direction, he will have to lay down his cigar with the rounded end pointing in the other direction. Otherwise, it is not properly mirrored and player two will be able to introduce asymmetries that will make the outcome of the game unpredictable. He could lay a cigar at an angle to the previous cigar, cutting off the small space allowed by the rounded shape of the end, which could not be duplicated in reverse unless the cigar itself had been placed in reverse as I described. If it had not been so reversed, it is possible that a cigar could be so placed that it would cut off the space for another cigar, by spanning two cigars for example."

"I see," said PJ. "The cigar must be symmetrical but reversed. Round end north, then next cigar placed on the diagonally opposite side with round end south."

"That would cover it," said Tollie.

Mouse hummed a little. He was proud of himself, for he had in fact seen the solution before anyone else, but pride was unbecoming in a Mouse, so he refrained from his little dance. In fact, only Tollie noticed the humming. PJ was already looking at them with a mischievous gleam in his eye. Mouse and Tollie could tell that another puzzler was coming...

### CHAPTER 21

PING-PONG BALL... AND PINS GALORE

PJ Nagrom paced for a while, chomping on one of his cigars. Tollie wondered if he ever really smoked them.

"Here, here," he said, "what you have is a ping-pong ball." PJ fell silent, as if this were enough.

Tollie and Mouse watched him. If there was a train of thought going on, they did not want to interrupt it.

"You have also a length of pipe, fastened to the floor at one end, the open end facing up, one foot high. The diameter is just such that you can drop a ping-pong ball into the pipe and it will rest on the floor. You can visualize this?

Mouse and Tollie nodded duly and affirmatively.

"Well, then, you also have a sharp pin such as a tailor uses to fasten hems, a baker's rolling pin, and a bowling pin, three various pins in all. How can you get the ping-pong ball out of the pipe?"

PJ Nagrom looked around at them triumphantly. He was sure this was the puzzle that would stump them.

Tollie indeed felt stumped! "I can see that you could perhaps spear the ping-pong ball with the sharp pin, but how would you get it down there? The rolling pin and the bowling pin don't seem to help."

Mouse looked dejected. He could not see any use for the pins at all. "If you used the sharp pin to spear the ping-pong ball, the ball would be damaged. This violates the premise that you are to get the ping-pong ball out of the pipe intact. I know it is not stated outright, but certainly there would be no use in ending up with anything less than a usable ping-pong ball. Suppose you had to crush the ball to get it out. That would seem to violate basic principles. You would not really be getting the ping-pong ball out of the pipe fairly, and we have only been working with fair problems, that have fair solutions."

"Well said!" Tollie agreed, and they both looked at PJ.

"I might be heavy, but I am not unfair!" protested PJ Nagrom. "You can get the ping-pong ball out intact and with no piercings or crinkles of any kind marring its spherical perfection." He huffed and chomped vigorously on his cigar, evidently offended at even a mere suggestion that he might not be posing a fully honest problem.

"Okay, okay, we did not mean to suggest that you were not being aboveboard or that your problem is not a legitimate one, with a legitimate solution," Tollie said to mollify the big man's displeasure. "We just cannot see how the pins can be of any help in getting the ping-pong ball out of the pipe!"

"I never said they would be," PJ said airily. "All I said was that you have them." He looked around at the ceiling, perhaps seeing things that were beyond the perceptive abilities of Tollie and Mouse to detect.

Said Mr Mouse at this point, "You say then that the pins might not have anything to do with the removal of the ping-pong ball from the pipe?"

"I have presented you with the problem at hand. It can be solved."

"If the pins are of no use, why did you mention them?" Tollie asked.

"Because, of course, you do have them, as the challenge is given."

"If we have to use other means to remove the ball," Tollie protested further, "surely you have to tell us what other means are available for us?"

"Not necessarily," PJ answered mysteriously. "There are some things that you might be said to have about you all the time." He looked at them, his posture challenging, arms folded, as though daring them to claim that this statement was not self-evident.

"I can think of no way," Mouse announced after a long pause, during which all had fallen into a meditative silence.

Tollie's scrunched-up face showed no sign of impending illumination of the solution to their newest challenge.

PJ waited patiently, seeming to believe they could indeed solve his puzzle, if only they had the right flash of insight.

But the silence dragged on, and PJ, Mouse, and Tollie all paced around in deep thought. Tollie was beginning to think that PJ's thinking was somehow mistaken. He could not bring himself to believe PJ was being deceptive, as it did not seem in line with what they had learned of his character generally.

But, finally, both he and Mouse had to acknowledge they could not find a way to get that ping-pong ball from the ground to the top opening of the pipe.

PJ laughed uproariously, his entire bulk quivering with mirth.

"Why," he said, "you just pee in the pipe and the ping-pong ball will float to the top."

The two traveling companions looked at him questioningly, plainly feeling there was something unfair about this answer, as it did not involve the use of even one of the various pins posed in the original challenge.

Seeing their injured expressions, PJ relented and offered an explanation. "This is an example of a puzzle that includes unnecessary information, things you do not need for the solution, that are only mentioned to mislead. That part is true. What is not true is the idea that such a puzzle is not completely honest. Like all fair puzzles, it has a fair solution!"

After a short pause, Tollie had a reply. "Yes, I can see that and I can grant you that it does have a solution, though you did mislead us. However, every word you said was true, so I will allow that your challenge was fair."

"But not as fair as our challenges have been so far," Mouse groused.

Tollie was surprised. He had never seen a grumpy side to Mouse. But, he figured, not even the inimitable Mr Mouse was completely perfect.

PJ seemed aggrieved. "I introduced a new dimension to your experience of puzzles. There are others, this is not the only direction in which one can take puzzles, to make them just that much more interesting. You will admit that without the element of surprise, puzzles would be in danger of becoming boring?"

"Yes, yes," Mouse chimed in, in a new tone of appreciation. "If we did not surprise now and then, life itself would become boring. You are right."

PJ bowed deeply with a flourish. "I like variation in life, as you say. That is why I enjoyed the cigar game puzzle so much. But, this one was different, I will agree with you on that. Forgive me, though, for it was different for a reason."

Tollie and Mouse both laughed. "I agree, PJ," said Mouse. "You did surprise me, and I did enjoy it after all."

Tollie hummed a little, happy that all was revealed and it made sense, and it had the advantage of teaching them a new idea.

He realized, too, that it was time for him and Mouse to leave. They had spent a whole day at the house of the three doctors! Tollie realized that the passage of time was different in the World-Wide Web. He did not feel like more than an hour or so had passed, but in fact, in this world, it had been a full day. His body was used to the passage of real time, so he had not felt the need for sleep. And Mouse probably never slept, having so much to do and so many places to go.

But there it was, a day later and time for them to take their leave of this fascinating house and head on back down the hill. All in this world happened on cue, and sure enough the three doctors, Mr Quirm, Li'l Jumbo, and PJ Nagrom were all there now to say their farewells to their guests. Tollie and Mouse made a graceful exit, thanking all and looking back and waving as they reached the head of the path that was to lead them back down the hill.

### Chapter 22

SAME PLACE, SAME TIME

"It is noon now, the same time as we started up the hill yesterday," Tollie noted to Mouse.

"So it is," Mouse agreed. "Exactly 24 hours that we spent. At this same time yesterday we were beginning our journey up the hill. Today at the same time we are heading back down. What a coincidence!"

"A whole different day, to the second. I wonder what we will encounter next."

"We shall see."

They wound their way down the path, noting afresh the beauty of the landscaping, a product of the imaginations of the three doctors that impressed them yet once more. It seemed different on the way down, though it was yet familiar. It had not changed, but took on a different aspect traversing it in the opposite direction. Impressive indeed!

Part of the effect was due to the many cascading waterfalls, which they now saw from above rather than below. Instead of water flowing toward them, it was now flowing away from them, falling over the edges smoothly, while on the way up, they had seen it falling off the edges and cascading down irregularly into their receiving pools. They wended their way down, taking their time to enjoy the carefully arranged landscaping. In time they arrived at the bench where they had rested on the way up.

"Odd," said Tollie, noting the time again. "We were at this place at precisely this time yesterday!"

"Not so odd," Mouse observed. "In fact, it is inevitable."

"I don't see how." Tollie considered Mouse's calm statement, with all its mystical, even fatalistic overtone. "We would not have walked at exactly the same pace, taken the same pauses, or even walked the same exact path. It is not at all clear to me that it is inevitable that we would arrive at the same point of the journey at the same time as yesterday."

"We started up at the same time yesterday as we started down today," Mouse suggested. "At some point on the journey down we would arrive at exactly the same location at exactly the same time as we did on the journey up."

Tollie considered this skeptically.

"Well, consider it a clue," Mouse added.

"Your clues have always been good ones, so there must be something I am missing. And, it must have to do with our starting out at the same time."

"Yes, indeed." Mouse took a seat on the bench and enjoyed the view below. He had not on the way up as he had then been admiring the house on the top of the hill. He looked so comfortable that Tollie yielded to the temptation and sat down as well, contemplating the island below and the great blue ocean stretching from its beaches. The sun-drenched vista induced in him a pleasant drowsiness, the first time he had felt any tiredness at all. Mouse smiled contentedly and enjoyed the moment. He was a hard-working creature and appreciated the rare chance that had been granted to him here and now to have nothing immediate that needed to be done.

"We started up yesterday at noon," Tollie mused out loud, proceeding in relaxed fashion. There was no hurry here. "We started down today at noon. After a certain amount of time we arrived today at this same location where we were at exactly this same time yesterday..."

He left it hanging, thinking about something that would not crystallize in his mind. But there was something there. Something about going up and then going down, the departures at the same time, their crossing paths with each other at this precise place...

He had it in a flash. Mouse could see it,, and said, "Good, you have thought it through as far as you could and then you had a sudden insight. Those are the most fun solutions."

"Of course, Mr Mouse," Tollie agreed, pleased with the result of his ponderings. "Suppose we made the trip in one day, only I started at the top, going down, and you started at the bottom, going up. Sooner or later we would cross paths, where we would be at the same point at the same moment in time. Since we took the same path, we would have to cross each other in only one place."

"That's right," Mouse confirmed. "What we did is the same thing, except we started up at noon one day and down at noon the next day. We would have to cross our previous path at the same point at the same time, just as if we made the trip separately on the same day and actually physically passed each other."

"How simple it is!" Tollie marveled that he had not at once seen it. It is inevitable, as you said.

"But I have had occasion many times to observe just this phenomenon," Mouse explained, "for I spend much of my time taking trips one way and then back again to the point of origin, very often through the very same series of locations each way. It occurred to me that if I had started back at the same time the next day as I had started out, I would be at the same location at the same time on the way back. Though I could not predict which location, there would have to be such a location. I would have to 'pass myself,' so to speak, as we have just now done."

They lapsed into silence, captivated once again by the surrounding beauty. This did not last long, though, for an otherworldly voice presently made itself heard. The voice reverberated through their world, clearly heard by them and yet not emanating from anything that they could encounter in the World-Wide Web. How Tollie knew this he could not say, though it was a fact that the voice was calling to him.

It was a familiar voice, as it well should have been. He had heard it all his life.

### CHAPTER 23

FAREWELL, MOUSE

"Tollie... Oooh, Tollie..."

It was his mother's voice, but he could not see her at once.

"Time for bed, Tollie."

"That's mom calling me," he told Mouse. "I have to go, it's bedtime."

"You will be back many times, Tollie," Mouse answered. "And we will explore many different things. It will be very exciting. I had a great time with you, it was an interesting excursion for me in my world. We went about pretty much where things took us, which was satisfying. Often I am directed here and there, whether or not I want to go. But that is my job. You gave me a good go of it."

"Thank you, Mr Mouse. I had a wondrous time myself."

Tollie was inspired to break into their World-Wide Web song and dance.

"We are, living in a dream,

living in a dream, dream,

streaming dream..."

Mouse joined him, dancing and singing hand in hand before taking leave of each other.

"Happy, happy are the mice.

happy are the nice, nice,

nice, nice mice..."

All the varied characters they had encountered in their travels showed up to bid Tollie a good night and many happy returns, and they joined in the dance, each singing their own song and doing their own dance, yet forming a great interconnected circle just the same.

Tollie made up a new little song for his farewell to Mouse.

"Once, yes, once upon a time,

'twas a wondrous, wondrous,

wondrous time..."

He had to leave it at that, as a deep whooshing sound transported him through a swirling maze of zips, zings and zwiggles, flashing lights and gossamer wires, depositing him squarely in his bedroom in front of his brand-new shining computer...

Mouse's faint voice followed him! "Tollie...

"How many letters are in the answer to this riddle?"

It was Mouse's farewell to him, Tollie realized. "What riddle?" he asked. But there was no answer. He was no longer in Mouse's world.

He realized that Mouse had left him a challenge to take with him. Could he solve it without Mouse there to give him a hint?

He would think about it tomorrow. It was not usual or characteristic of him, but now he was truly ready for bedtime – he who always pleaded to stay up just a little later. He looked at the clock and saw that he had already been allowed a late night!

Laying down, he mulled over just how many letters were in the answer to this riddle. And fell asleep. Tomorrow he would solve Mouse's farewell offering...

It was a wonderful birthday!

