MALE SPEAKER: Well, hello,
welcome to a Google Tech Talk
called "Martin Gardner 101."
Our speaker is
Colm Mulcahy, who's
in the math
department at Spelman.
He had the pleasure of
knowing Martin Gardner
during the last decade
of Martin Gardner's life.
Martin Gardner died in 2010
and would be 101 this year,
which is part of why this is
called "Martin Gardner 101."
Colm has had puzzles published
in "The New York Times"
and he's written for
"Scientific American."
COLM MULCAHY: So thank
you very much, Rick.
Great pleasure to be here.
Thanks for coming out,
taking a break during lunch.
I hope to tell you something
about this amazing man, Martin
Gardner, and some of the
puzzles he came up with.
A lot more can be found online.
He published an enormous
amount of material.
I'm going to go over maybe
six puzzles related to him
that are not so well-known.
One or two of them
are well-known
and perhaps the rest
are not so well-known.
Some open problems too.
He asked questions that we
still don't know the answer to,
which is quite extraordinary.
So as Rick said, I'm based at
Spelman College in Atlanta,
Georgia, and I go by
the name Card Colm
because I write a column
every two months for the MAA,
"Mathematical Card Magic."
And that's all freely
available online.
MALE SPEAKER: Want to
do the web browser?
COLM MULCAHY: Oh, yeah.
Let's flip back to that.
So if you Google
Conway's Game of Life,
you-- it's hard to see it
here, but actually what's
percolating up in the
top right-hand corner,
it's very washed out
there, is a simulation
or a real-time reenactment
of the Game of Life.
So it's an Easter egg that some
clever person here [LAUGHS]
has put in.
So the Game of Life was
a cellular automata thing
that he wrote about, that Martin
Gardner wrote about in 1970
and it became hugely popular.
So it's nice that when
you Google the term,
you automatically get to
learn a little bit about it.
Thank you.
So I call Martin Gardner the
best friend mathematics ever
had.
He was probably
also the best friend
that rationality ever had.
And I just came from the
amazing meeting in Las Vegas
where Randy and various
other people of that ilk
were talking about rationality
and debunking pseudoscience,
and Martin turned out to be a
major player in that very early
on.
He was born in 1914,
21st of October--
same day as Dizzy Gillespie
but a couple years apart.
And last year, of course,
he would have turned 100.
Now, I took this
photograph of him
the first time I
went to visit him.
He lived in retirement
in Oklahoma,
and he was a very shy man.
And as soon as he saw
me taking my camera out,
he really just froze up.
He just didn't like having his
picture taken, very reluctant.
But I said, Martin,
please, I want
you to stand in front
of this bookcase,
and I want to take this picture.
And it just happened to be one
of the better pictures of him.
He doesn't quite
so grumpy as usual.
When the camera's out, he
was not grumpy in real life.
And it was used by "The New York
Times," and "Time Magazine,"
and so on when he passed
in their obituaries.
I call it Martin
Gardner standing
by every word he ever wrote.
[LAUGHTER] Plus, he actually
wrote all these books,
and you can see
four shelves' worth,
and there were two
more below the belt.
He actually wrote an enormous
amount, a very productive life.
He wrote over 101 books.
In fact, he wrote 101 nonfiction
books and about three fiction
books as well.
So, you know, when
he died, people
said he wrote 60 or 70 books.
That's kind of true in
the mathematical sense.
But, you know,
when my cat dies, I
don't want the obituaries to
say, this cat had three legs.
My cat has four
legs, and it deserves
credit for the fourth leg.
Martin Gardner wrote
104 books, and he
should get credit for that.
I've got a list of them online.
And mostly on math,
physics puzzles.
He wrote two famous books
on physics, philosophy,
and all sorts of other stuff.
Most well-known, of course,
for "Scientific American."
He wrote for
"Scientific American"
from the mid '50s to the
early '80s continuously
and then sporadically
for a few more years.
And that gave rise to about
300 columns, many of which
are very, very influential.
You can find those in 15 books.
Rick had one earlier.
The "Knotted Doughnuts" one,
I think, is doing the rounds.
But the nice thing is they're
all connected in one place
as well in a searchable
CD-ROM by the Math
Association of America.
And that's cool
because if you want
to look up magic squares
or Persi Diaconis,
you don't have to leaf
through the index of 15
different books.
You can just find
it very quickly.
And it's about $40 or
$50-- highly recommend it.
Every library should have one.
Every teenager should have one.
Actually, Knuth is
quoted back there
as saying that this
is "The Canon,"
and I've discovered sometimes
that some people say, well,
who's Knuth?
In fact-- [LAUGHTER] he was
asked this question himself
four years ago.
He's a professor at Stanford,
retired professor at Stanford.
And he was asked, well, younger
computer science students
may not know who you are.
Do you think that's bad?
And I find his answer
was very interesting,
because his answer was,
he was far more upset
that people might not know
who Martin Gardner was.
So he deflected the
question entirely,
and this was about a year
after Martin had died.
He said Martin did a lot
to illustrate mathematics
in an interesting way.
And he says, for me,
the important thing is
don't remember me, but
I did write these books
and I got things right, and
don't reinvent the same things
over and over.
So when you're reading
"The Art of Programming"
or if you're not
reading it, you probably
should be reading it,
rather than redesigning
the same sorting algorithms.
Sorry.
So among the topics that Martin
made famous and depending
on where you are on the
age spectrum, some of these
may be familiar.
He started off with something
called hexflexagons.
That was a folding activity
with strips of paper
that were folded in
hexagonal patterns
and funny stuff happened.
And that was actually
first explored
in [INAUDIBLE] by a team of
young students in Princeton,
including Richard Feynman.
And Martin
resurrected this topic
and wrote about it for
"Scientific American"
in 1956, the very end of '56.
It was so successful just
the buzz even in the office,
in the "Scientific
American" office,
that before it even went to
press, it hit the streets,
they said, this has
really taken off.
Would you like to do
something like this regularly?
Do you think there'd be enough
material to do a regular column
on mathematical recreations?
Martin lied through his teeth
and said, oh, of course.
[LAUGHTER] And then rushed
down to the nearest bookshop
he could find and
get all books that
then existed on
recreational math
and started writing about it.
Started off with simple topics
like magic squares, but as time
went by, the stuff
got more sophisticated
because he started getting
friendly with people
like John Conway, and Knuth,
and Diaconis, some of whom
were students at the
time and some of whom
were already professionals.
He met Persi Diaconis,
the famous magician
and statistician, in 1959
when Persi was only 13,
and they were friends
for a very long time.
So another thing he wrote
about was the Soma cube,
which is basically a three by
three by three jigsaw puzzle.
It's a three-dimensional
jigsaw puzzle
with 27 cubes that have been
preluded to seven shapes
and you have to
assemble them together.
Conway as an
undergraduate figured out
with Richard Guy's son
Mike Guy, that there
were about 2,000 different
ways you could do that,
and he did it
using group theory.
So right away, Martin would
write about something,
and mathematicians who
were either already famous
or budding mathematicians
would take it, run with it,
and do something
interesting-- you know,
classify, generalize,
find the patterns.
The art of Escher,
M.C. Escher's art,
which adorned many
a bedroom wall
in the '60s, and '70s, and
'80s-- that became well-known
because of Martin Gardner.
Outside of Holland, not
well-known at the time.
And Martin wrote an
article about him,
which we'll have more to
say about later, that helped
popularize Escher's work.
Lots of things in here.
I showed you the Conway Game of
Life briefly, the Easter egg.
When you Google that,
you find fun stuff.
Cryptography, there's a story
that Martin broke in 1977
or so.
RSA cryptography was
broken by Martin.
Martin knew the guys
who invented it,
they got in touch with
him, and he published it.
And then there was
an issue of should
this have been published.
And there were some people in
suits who came to his office
and said, cease and desist,
and don't send the paper out,
but the cat was out of
the bag at that point,
so the story became well-known.
Fractals-- fractals became
known to the general public
because it was a cover story
in "Scientific American"
that Martin Gardner wrote.
He was a journalist, and he was
a science journalist, science
writer, and he just happened
to be there at the right time.
Mandelbrot was one
of his neighbors,
so he got the scoop on
fractals because the guy
lived a few blocks away.
Paradoxes-- when
I was a teenager,
I remember reading some of
the Martin Gardner books
and being fascinated
with the paradoxes.
The all crows are
black paradoxes,
the paradoxes of infinity.
Some infinities are
bigger than others.
Big surprise when you're 16
or 17, but it's very true.
And Martin wrote
that kind of thing
and Fermat's last theorem
to the general public
before anybody else did.
And then Newcomb's
paradox, which
is still unresolved, surprising
things like nontransitive dice.
I can present you
with three dice
and say, which one
would you like?
You pick one, I pick one.
My roll would beat
your roll on average,
no matter which one you pick.
A is bigger than C is
bigger than A. A is bigger
than B, is bigger than
C, is bigger than A.
So very surprising.
This was actually
the picture that
was used on the cover of the
first paperback of a Martin
Gardner book I had
when I was a teenager,
and the question was, how many
pieces at most if you're clever
can you slice a bagel or a donut
into with three planar cuts?
And the answer is not
obvious, and in fact, you
can't tell from this picture.
You can kind of get a lower
bound to maybe an upper bound,
but the actual answer I
think is 13 for three cuts.
But nobody had asked that
kind of question before.
Anthropologists didn't have
the ready answer and now,
of course, we know the answer
for any number of cuts.
But this was cutting-edge stuff
back in the '50s and '60s,
and you can try it at home
with a bagel or a donut.
So math research
can be achieved.
The overhead is not high.
Now, he wrote all these columns
for "Scientific American,"
so last year for
his centenary, we
got carried away with
this top 10 theme.
And I wrote a piece for
"Scientific American"
called "The Top 10 Martin
Gardner 'Scientific American'
Articles."
There was also the top
10 Martin Gardner books,
the top 10 Martin Gardner
magic tricks, the top 10 Martin
Gardner physics puzzles.
So if you Google
Martin Gardner top 10,
you will find all of these.
There's about-- in fact,
it got to the point
where at the end of
the month, at the end
of the centenary
month last October,
we had to come up with a
top 10 Martin Gardner top 10
list-- [LAUGHTER] because
there were about 13 contenders.
So here's the list
of the famous,
the really famous top 10 columns
for "Scientific American."
The flexagons,
complex dominoes--
that's polyominoes
and pentominoes and so
on-- the geometry one there
is Coxeter's geometry,
hyperbolic geometry, and that's
where the Escher art came in.
I skipped over number three.
Every spring, he would
put out a column which
just had nine short puzzles,
and those are the things
that a lot of people
remember him for.
Great brain teasers
and then the solution
would come a month or two later.
And the interesting thing was
the solution would come a month
or two later, but sometimes
in that month, the solution
he intended to present changed,
because people would send him
new solutions that
he hadn't thought of
and that the person who
had given the problem
hadn't thought of.
So, and then sometimes,
people would write in and say
two months later, oh, but
you forgot about this case,
and we can generalize this.
So it was this constant
feedback between him
and his audience with a kind
of a month or two delay.
And you know, if you
were interested in doing
these problems back in the '50s,
and '60s, and '70s, and '80s,
you had to do them and
wait a month or two,
because there was no Google
to look up the answer, right.
So you had to do it
yourself or find somebody
who could help you out.
So number five there
is the Game of Life,
and that came out
in October 1970.
And Conway's biography
just came out,
and his biographer
claims in there
that there's a very substantial
claim that one-quarter
of the world's mainframes
were playing the Game of Life
in the year 1971, 1972.
Not that many people have
access to computers back then,
of course.
The free will paradox, the
Newcomb paradox is number six.
Number seven's great.
It was an April Fool's gag,
April Fool 40 years ago,
1975, April Fool.
So we had six or
seven things in there
that he claimed had
been overlooked.
And again, if you read them,
you couldn't check easily
because there were
libraries and encyclopedias,
but they were printed
several years earlier.
So new results that
somebody announced it,
people tended to believe you.
You couldn't check
by looking it up
to see if it was a hoax, right.
So among the things in there
was a claim that Leonardo da
Vinci had invented the
valve flush toilet,
and he backed it up with
some very impressive artwork
that a friend of
his did, somebody
who illustrated many
of his books and he
claimed he'd found in the
New York Public Library.
So it looked kind of genuine.
He also had a perpetual
motion machine,
but that was invented
by a Mrs. Bird Brain.
[LAUGHTER] So if you were
actually paying attention,
you probably should have
got that it was a joke.
And he had some math
ones in there too.
The curious thing is
that column generated
a huge amount of response,
but interestingly,
a lot of the response
was, I love that article.
Five of those discoveries
were absolutely fascinating,
but I think you made a
mistake on the sixth one.
[LAUGHTER] And
which one the reader
thought was a mistake was
kind of uniformly distributed.
[LAUGHTER] Like roll the die.
The same thing happened actually
with the skepticism books.
When he first wrote about L. Ron
Hubbard, and dowsing, and UFOs,
and astrology, people
would write in and say,
I love that book, n minus one
of the chapters are great,
but you really messed
up on this one.
So that was kind of
a recurring theme
that when he was not quite
serious about something
or poking fun at
something, people
tended to only get insulted
if it was about them.
They thought everything
else was fine.
By the way, for this April,
we did an April Fool's joke
with "Scientific American."
We got John Conway to
regale us with tales
of Martin Gardner writing
a joint paper with Erdos
with Nicholai Bourbaki, a
fictitious mathematician.
But it's all on video
so-- it's on the internet
so it must be true, right.
So that's April this year.
At number eight
there is fractals.
Number nine is Penrose
tiles, and number 10
is RSA cryptography.
So these are 10
astonishing stories.
Any journalist would
give their right arm
to be able to introduce the
public to any two or three
of these, and Martin luckily got
all ten and a lot more besides.
So those are, I think,
the most famous ones.
He also was fortunate in
that his work was featured
on the cover or
artist's work related
to the things he
was writing about
were featured on the cover.
And there were
actually 10 or 12,
I should say-- 12 cover
stories, and they're not
correlated exactly with the
top 10 ones I just showed you.
But you can see here we have
down at the bottom right,
we have fractals.
To the left of that,
we got Penrose tiles.
The top left-hand
corner's a great one.
It's like decomposing a
square into smaller squares.
Can it be done or not?
That-- if you actually examined
that with a magnifying glass,
it's not quite a square.
And it was an open problem when
he wrote about it in the '50s,
but it was solved in the '70s
I think on a much bigger scale.
Somebody found squares that
fit together in the-- no,
not like the floor
over here and here,
but in a the surprising way.
The third image
here from the top
is the Penrose-- sorry,
the work of Escher.
What happened with Escher was
Escher liked Martin's stuff,
and Escher wrote him a fan
letter from Holland in 1960.
And he said, I'm a great
admirer of your work,
blah, blah, blah--
by the way, would you
like to see some of my etchings?
[LAUGHTER] And he slipped
in a couple of things,
and Martin looked
at those and went,
wow, and got in touch
with Coxeter in Toronto
and hence this article which
came out in spring of '61.
And notice how there's
colors in there.
It's geese flying
in one direction,
white geese versus colored geese
flying in the other direction.
Escher was outraged that
"Scientific American" artist
had seen fit to color
his work, because it
was originally black and white.
He got over his outrage as
the checks started coming in.
[LAUGHTER] And sadly,
for Escher, he only-- he
died in '72 so most of his
fame was after his death.
But Martin was the first
one to break that story.
OK, so those are
some of the covers.
Now, as the legacy
of Martin, here
are some interesting
quotes from Knuth again.
Knuth says more
people have probably
learned more good mathematical
ideas from Gardner
than from any other person
in the history of the world.
That's an astonishing claim.
I mean, what contenders
could there be?
Euclid's elements,
which may or may not
have been written by Euclid.
It certainly has been revamped
over the centuries, millennia
since.
Jim Propp has a
great quote that says
that Martin Gardner's
column was the best watering
hole of its day and
beside-- behind the scenes,
he was a tireless matchmaker.
That's a very important and
relevant comment, I think.
Simon Singh, you may know him,
he's a UK-based journalist.
He's just done some
book tours over here,
says that he hopes that future
generations of budding nerds
will be introduced to
Martin Gardner, the greatest
popularized mathematics of the
20th century, if not all time,
and that they also be
introduced to his writings
on skepticism and science.
See, Martin was a
shy guy, and one
of the bad downsides of that is
that he never had a web page.
He never promoted himself.
He just wrote books and moved
on to write the next book.
He never had the profile of some
other people in related arenas.
He was no Steve Jobs,
he was no Carl Sagan,
he was no Richard
Feynman, but yet, I
think he belongs up there
with those three guys.
And we'd like your help
to make that happen.
And then Ron Graham again,
who's a mathematician who's
worked with Persi Diaconis
a lot and has known Martin
for 50 years, said that
there's no question that he's
responsible for turning
people of all ages
onto math more
than anybody else,
and nobody else has succeeded.
People always say, who's the
next Martin Gardner, who writes
as well as Martin Gardner?
Nobody, it's that simple.
While people have
tried, but Martin-- see,
it was a full-time
job for him, which
was interesting because the rest
of us tend to have day jobs.
We teach, and grade papers,
and program, or whatever we do.
Martin didn't have to do any of
that, and that really helped.
So I think he was a
great community builder.
He built a huge community, and
we have harvested testimonials
from people over a huge
age range-- generation
after generation of
people have he influenced.
He wrote reviews and books
right up to his death.
I mean, a day or
two before he died,
he was still writing stuff.
He processed information
at a ferocious rate.
He read a lot, he
thought deeply.
He wrote, moved onto
the next project.
There are several
generations of people
influenced by him and of
course gatherings in his honor.
There's a gathering every
two years in Atlanta
called the Gathering
for Gardner, which
is a by invitation only
conference of mathematicians,
puzzle makers-- I mean, physical
puzzles too-- magicians,
skeptics, and so on.
If you're interested in
that, let me know afterwards.
And then every October,
we'll say more about it
later, we invite
the world to join
in celebrating the kind of
things that Martin liked.
It's not about him,
it's about the ideas,
you know, making people
think rationally,
and making people
curious, and making
people expand their brains.
But interestingly,
people say, well, it's
just silly, frivolous
recreational mathematics.
No, it's not, and I
hope to convince you
of that in the
time that remains,
because new results
of substance have been
and continue to be produced
as a result of questions
Martin asked and
columns that he wrote.
He was a firm believer
that recreational math
was a foot in the
door to real research
and in fact, formed the natural
springboard for curiosity
and innovation.
And I think Strick
mentioned in the flyer
that he circulated yesterday
that there was a time when
some of the jobs questions,
interview questions you might
be asked at a
company like Google
or perhaps other companies,
Microsoft, some of those
came from the kind of puzzles
that Martin popularized.
I'll tell you
something else that
came from it, uncredited for
most of his life-- "Car Talk."
"Car Talk" frequently introduced
their shows with a puzzle,
and they nicked them
from Martin frequently.
And it's funny because they
got a bee in their bonnet
about 10 years ago for
reasons we don't understand.
They thought he had died.
Apparently they didn't
Google check. [LAUGHTER]
And they announced on
there that Martin Gardner
had died and
confessed that they'd
been stealing his puzzles
for all these years.
Martin Gardner's son was
listening-- [LAUGHTER] And he
had talked to his father
like maybe the day before,
so he got a little worried.
He picked up the
phone, called dad.
Dad, are you OK?
I'm fine, what's going on?
I'll call you back. [LAUGHTER]
And then he called NPR,
and he said, I live in
Oklahoma, and I know this man
and he's alive.
And how well do you know him?
Well, you know, he's
my dad, so-- [LAUGHTER]
So then they did an apology
broadcast the next week,
and when Martin really died,
they rebroadcast all of that.
And you can-- it used to
be free to listen to it.
Now it costs a dollar.
It's June 2010-- one of the
June podcasts of "Car Talk."
You can listen to it, it's fun.
They also have a CD out
called "The Best of 25 Years
of 'Car Talk'" and Martin
Gardner gets a track on that.
It's one of their greatest
hits, so there you go.
So I said that Martin was a
firm believer in math forming
a springboard for curiosity and
also can lead to real research.
In fact, there's kind of
two sides of this coin,
and these are direct
quotes from Martin.
He says the best way to
wake up somebody to math
is to introduce a game, a
puzzle, a magic trick, a joke
even, or a model,
or a paradox, or any
of a score of other things that
dull teachers tend to avoid
because they seem frivolous.
And as a teacher myself, I'm a
math teacher at college level,
you often think, well,
the test is coming up.
I've got to get
through these theories.
I don't have time to
introduce the fun stuff.
But I've talked to
a number of teachers
who say, take the time,
take five minutes out.
Maybe start a class every Friday
to do--to introduce a Martin
Gardner puzzle, because
students look forward to it.
They wake up, and their
brains work better
for the rest of the class
even though they're going back
to their normal work.
But more importantly,
Martin says
that you'd be surprised how much
math you can learn by exploring
the implications and
ramifications of something that
just seems like a brain teaser.
And John Conway is
another living example
of both of these.
John Conway invents
games constantly.
And in fact, I was just
reading his biography,
and it said that he
invented the Game of Life
in an attempt to outwit people
who were using computers
to solve math problems.
He's like, I'll give them
something a computer won't
be able to solve, and that
led to a whole subculture
of its own that's been
going on for 45 years now.
In fact, John Conway--
one of his famous quotes
is, in all caps, I
hate the game of life.
It's the thing he's
most famous for,
and that bothers him
because he's obviously
done a lot of other things.
But Martin was the one who
used the midwife analogous.
So the testimonials.
So if you go to the
martin-gardner.org site,
on the top left, there's a
link called testimonials.
And we're collecting
stuff there.
We've actually got--
we're almost at 100.
We'd like to get to 100 because
we're in the centennial year.
And the age span varies.
Well, Richard Guy
contributed yesterday.
Richard Guy is the
subject with Conway
and Elwyn Berlekamp from
his neck of the woods.
They're the subject of a
big conference coming up
this Monday and Tuesday in New
York at the Museum of Math,
the Moves Conference.
Richard Guy's still doing
stuff, a lot of stuff.
And in September, he turns 99.
That's astonishing.
I sent an email out
yesterday asking
for some new testimonials.
He got back to me in
about three minutes.
He was faster than all the
70-year-olds and 80-year-olds.
[LAUGHTER]
But we also had Ethan Brown.
If you don't know
who Ethan Brown is,
he's worth looking up.
He's a teenager who is
astounding, talented,
and I think we're going to
be hearing more from him.
So we have an 80-year age
span, and we'd like yours.
A lot of the people we have in
here-- Hannah Fry from the UK,
James Randi, of course,
Conway himself, Richard Guy,
Simon Singh.
Stan Isaacs is
based in this area.
He is the man who curated
the Martin Gardner
archives which are at Stanford.
So Ray Smullyan, the man
who wrote the book, "What
is the Name of This Book?"
Back in 1978, that was funny.
Now we're jaded,
but 1978, that was
revolutionary to publish
a book called, "What
is the Name of This Book?"
Raymond Smullyan's about
96, and he's still rocking.
Mike Reiss-- one of the
"Simpsons" creators.
A lot of people have left
us some cool comments here.
Gary Antonick, "New York
Times," is with us today.
Rudy Rucker, Ethan
Brown's in there,
Alvy Ray Smith, one
of the Pixar founders.
People from all sorts of
different areas, because Martin
didn't just appeal to
one side of the brain
or to science people.
In fact, among his
associates and friends
were Isaac Asimov
and Salvador Dali.
But that's not what we
need to mention today.
Now, because last
year was a centennial,
we managed to get the people
who run math awareness month
to make it in honor of Martin.
And that was April of last year.
So a new activity
went up every day
throughout the month of
April, but the good news
is they stay up for all time,
because this website will
be maintained.
If you go now, you
get this year's theme.
You've got to go slash
2014 to get to last year's.
What we did was we took
things that Martin had already
written about and
kind of updated them,
made them video, interactive,
multimedia-friendly,
had younger people
presenting them.
And also some things that
weren't known to Martin
because they happened
a little after his time
such as the connection
between [INAUDIBLE]
if you were at the
beginning and fractals.
There's an interesting
connection that's
only come to light recently.
It's in the book by
Diaconis and Graham.
So this book on the
left here is the picture
that's actually a famous book
of Martin's, first book relating
to math that he published
the same year he
started at "Scientific
American" back in 1956.
It's called "Mathematics,
Magic, & Mystery."
It's still in print
with the same cover.
And on the right is the poster
for math awareness month--
same title basically.
And there were 30 days in
April, which factors nicely
as 5 times 6, so it's
kind of like an advent
calendar for people who
believe in mathematics.
And the top left was
Ethan Brown telling you
how to do in your head,
on the fly, four by four
customized magic squares.
And he makes it sound easy.
So everything's introduced with
a video of like 90 seconds,
2 minutes tops, and then
they explained some math.
The goal of something a
teenager could understand up
to grad school level.
The second one's about braiding.
There's card tricks and there's
turning the sphere inside out.
There's an optical illusion.
They think you drag in all
sorts of fun stuff juggling.
So we recommend
you check that out.
He got very generous coverage
in his centennial year.
All the major math
magazines in the US
and some of the key physics
magazines for teachers,
"Physics World,"
and the "APS Physics
Teacher," "Skeptical Inquirer."
Of course, it turned
out he was one
of the founders of
that back in the '70s.
The "BBC Magazine"
had an article on him
which got half a million hits in
the first day, which was great.
So lots of feedback.
And here again are
some of the covers.
He made it onto the cover
of quite a few magazines,
including all four of the
American magic magazines.
So let's get to a puzzle.
I'm going to actually show you
six puzzles, and some are easy
and some are not.
This is not such an easy one.
So I'm going to
ask you this puzzle
and you can think about
it, and at the end,
I'm going to show you
silently the solution
and give you the opportunity
to take a power nap if you
don't want to see the solution.
And if you haven't
solved it, you probably
don't want to see it, because
it's really important you
solve this yourself.
So this is it.
So five years ago, just a couple
of months before Martin died,
Gary Foshee who makes
physical puzzles and things
stood up and gave this talk.
And I listened very
carefully, and he sat down.
He said 20 words
and he sat down,
and the room went nuts because
of what these 20 words are.
So here they are.
I have two children.
One is a boy born on a Tuesday.
What is the probability
I have two boys?
Most people respond
by saying, what's
choosing have to do with it?
How could choosing
possibly be relevant here?
OK, so you can think about that.
Now, there's a much
simpler version
that Martin popularized
back in the '60s.
Very simple, it's
standard now when
you're teaching probability.
And Mr. Smith has two children.
At least one of them is a boy.
What's the probability that
both children are boys?
Compare and contrast with
Mr. Jones has two children.
The older is the girl.
What's the probability that
both children are girls?
So you have to write
out your sample space
and do a little
conditional probability.
So if you can't solve the ones
at the bottom of this page,
you probably won't master
the one at the top.
But it's not the
other way around.
Just because you can solve
the one at the bottom
doesn't mean you will
nail the one at the top.
The one at the top
is a little harder,
so we'll come back
to it at the end.
But it's an
interesting question.
I have two children.
One is a boy born on a Tuesday.
By the way, Martin
Gardner has two children.
[LAUGHTER] And I happen to
know that one of them is a boy.
And I was sitting beside Martin
Gardner's son about a year
ago at Princeton, and across
the table was John Conway.
Now, John Conway is
famous among other things
for being able to
spit out the day
of the week for
any date in history
and any calendar system
you could mention.
So I said to Jim
Gardner, Martin's son,
when were you born?
And he told me it was some date
and some month in the 1950s.
And I turned to John Conway,
and I said, what day was that?
And he blurted out Wednesday.
And I thought, oh, well.
And he said, no, I'm an idiot.
I'm an idiot.
He was really embarrassed
because it was in fact Tuesday.
[LAUGHTER] He made a
mistake, but you know--
So Martin Gardner in
fact has two children,
and one is the boy
born on a Tuesday,
so that's something
else to think about.
This is a very famous classic.
Two missiles speed directly
towards each other.
One's moving at
9,000 miles per hour
and the other is moving
at 21,000 miles per hour.
Now, 9 and 21 add up
very conveniently to 30.
And that's suggestive
if you know anything
about relative velocity.
And then you're told they
start 1,317 miles apart.
That's an ugly number.
And there's a question.
And the question is, without
using pencil and paper,
calculate how far apart they are
one minute before they collide.
[LAUGHTER] Now, the formula
connecting the relevant things
here is a simple formula.
Average speed is distance
traveled over time taken,
so you might have to multiply,
you might have to divide.
And you get nervous,
right, because that 1,317
isn't so fun.
And you might be
able to multiply it
by something in your head,
but you certainly don't
want to divide something into
it or divide it into something.
So you're worried, right?
Now, he said without
pen and paper.
This question was
posed 50 years ago.
Calculators weren't common
yet, so there was no question
of using a calculator.
And you're not supposed to
use a slide rule either.
True story, Martin Gardner
to his death balanced
his checkbook with an abacus.
[LAUGHTER] You're not supposed
to use an abacus either.
OK, well, of course, you do
realize that relevant velocity
is what's important
here, so it's
as if one object is stationary
and the other is barreling
towards it at 30,000
miles per hour.
There's still that
unpleasant number there.
Well, wait a second.
It's actually much
easier than you think,
because distance is
speed times time taken.
And it's a minute, but that's--
a minute which is one 60th
of an hour.
So the answer pops
out at 500 miles.
And you're thinking, what
happened to the 1,317?
It wasn't necessary
to know where they--
how far apart they were
when this thing started.
It's good that it's
bigger than 500,
because if I said they
started 317 miles apart,
we'd have a problem,
Houston, OK.
But now, so is this a
dirty trick question?
That's an interesting
question, and as a teacher,
when I first saw this--
I mean, I probably
saw it when I was
young and forgotten,
but when I revisited it
in more recent decades,
I thought, well, yeah,
it is kind of-- would I
dare give that to my students?
I would actually suggest that
maybe we should give problems
like this to our students
a lot more than we do,
because here we have a
situation where there's
extraneous information,
information that wasn't
necessary to solve the problem.
If you leave that out to make
life easier for your students,
what are you
training them to do?
You're spoon-feeding them.
You're making it so you
always giving them exactly
the right bit of information.
Never give them too much,
because they'll complain.
Never give them too little,
because they'll complain even
louder that it was impossible.
When you look out the
window through the world,
we're bombarded with data.
You guys know a lot
about data, right.
There's data everywhere.
You have to learn how
to sift through it
and decide what's important,
what's not important.
So maybe we should
give students problems
like this frequently, like
once a semester or once
a week, once a class.
I think our students might
be better prepared if we did,
so I think there's actually
a moral behind this.
OK, here's an interesting one.
I'm going to show
you some cans I think
there's six of them from--
we're going to number
them from left to right.
And my question is, for which
of them is the height greater
than the circumference?
So on the left,
we have chickpeas.
Then we have Pringles
it looks like.
Then we got coffee, the coffee.
I think it's nuts, cat food,
something sporty, and pineapple
juice.
So the little cat
food one, surely it
can't be tall enough
that the height exceeds
the circumference, so
we'll exclude number five.
What about the chickpeas
and the pineapple juice
on the two extremes?
Do they look tall
enough that the height
exceeds the circumference?
Depends-- you know, some
people think yeah, maybe.
The coffee can I would
throw out too and the nuts.
How about if we focus on the two
apparently more extreme ones?
Number two and number six.
Are those tall enough?
To my eye right now and I've
looked at this for a few years,
it looks to me like
those two tall ones
are maybe one and a half times
as tall as the circumference.
Now, it's possible to figure
this out without measuring.
I mean, if I had cans here,
we could actually measure it,
because if you look at the
can that says Penn on it,
that's actually got
three tennis balls.
[LAUGHTER] Which means
that the height is
three times the width.
Now, the last time I checked,
the circumference of a circle
was pi times the width.
How many decimals of
pi do you need to know?
[LAUGHTER] None.
And that's why this question
was called three-point question
mark.
That's all you'll need to know.
The fact that pi is
three point something.
You don't even need
to know it's 3.1.
It's bigger than 3.
So in fact, none of these
cans are tall enough
that their height exceeds
the circumference.
It's a little surprising.
Now, I was inspired to come
up with that because the very
first day I met
Martin, which was 2006,
after a little
introductory chat,
he said I've got
something to show,
and he produced this
tall, skinny glass which
you can see here.
Whipped out my camera
because I thought
this might be worth capturing.
And he said, do you think
this glass is taller
than it is round?
And look at it.
It looks like it's hugely
tall compared to round.
So of course, I fell for
it hook line and sinker
and said, oh, yeah.
Without saying a word, he took
out a little ribbon of paper
and showed me the
error of my ways.
We're very bad at gauging girth.
We're just lousy at it.
We're also bad at gauging
depth, but girth we're
particularly bad at.
And you know, as you get
older, you look in the mirror,
you might be putting on a few
pounds and you're in denial.
You don't see yourself
putting on weight, right.
We just don't see girth at all.
So maybe that's because of
too much appreciation of pi,
certainly in my case.
Yeah, Martin was a
"Scientific American" hero,
for Scientific American, and
he was also a magical American.
In fact, he invented
close-up magic
and published it from
the age of 15 and 1/2
in the spring of 1930 until
95 and a 1/2 years later.
One of his last published,
very last published things
is also a magic trick.
He was a rationalist
and debunker.
In fact, even as a kid, he read
stuff, early science fiction
magazines that had articles
on astrology and UFOs
and so on back in the '20s.
I didn't know the stuff
went back that far,
so he was skeptical
from a very early age.
He founded the skeptical
movement in 1952
inadvertently by publishing a
book called "Fads and Fallacies
in the Name of Science."
That was the first
book of that type,
and now, it's happily
a large movement.
But actually, he's most
famous for something-- nothing
to do with anything
I've mentioned so far.
"The Annotated Alice in
Wonderland" is his biggest hit.
And that's why Escher wrote too.
Escher was a huge Alice fan.
He actually knew about
"Scientific American,"
but he loves "The
Annotated Alice,"
and that's what prompted
Escher to write to Martin
and include his etchings,
which helped him get color,
colorize them, little more
well-known in the west
outside of Holland.
And this year is the
200-- 150th anniversary
of the publication of "Alice."
It was the fall of
1865, and there's
going to be an 150th
anniversary edition
of "Alice" and a new
version, because Martin
left some notes behind for
updating "The Annotated Alice."
So the definitive,
definitive "Annotated Alice"
is coming out in a few months.
Now, Martin-- a lot
of people assume
that Martin because he
was such a rationalist
must've been an
atheist or agnostic.
And to be honest, I would
have been in the same camp
until I did my homework.
It turns out he wasn't, but
he didn't talk about it much.
He wrote about it, and if
people didn't pay attention,
he didn't really care.
He actually considered himself
to be a philosophical theist
and a mysterian.
So we're back to math,
magic, and mystery.
A lot of math, a lot of magic.
He did magic way before he did
math, and to his dying day,
he considered himself
to be a mysterian.
And you can look up what that
might mean if you don't know.
So, but there's also a
mystery about something else.
Given what I've
told you about him,
it would be reasonable
to assume this guy was
an academic, right.
Probably had a Bachelor's,
Master's, maybe a doctorate,
and he taught students and had
fun grading papers all weekend
like I do, and published
in refereed journals,
went around the
country giving talks.
No, that's not Martin at all.
He actually never took a
math class past high school.
That's astonishing.
He graduated high
school in 1932.
He would have learned a
little bit of algebra,
a little bit of trig, and
a lot of geometry-- a lot
more geometry than we
learn today, but that's it.
There was no AP Calc.
Probably didn't get
to calculus until they
were juniors in college.
He actually intended to go
to Caltech to study physics.
That was his goal.
He wanted to be a physicist,
but in those days,
you couldn't just
walk in as a teenager.
You had to go somewhere else and
prove yourself for two years,
like go to the-- not a community
college, but another place.
So he went to
University of Chicago
with the intention of
switching after a year or two,
but he fell in love
with philosophy.
And Russell and Whitehead
were among the guest lecturers
who came by that he saw speak.
He also saw Houdini do escapes.
I mean, he saw incredible names
from various parts of history.
So he did have one university
degree from Chicago in 1936,
but it was in philosophy.
He didn't take any
math in college,
but he was interested in magic.
I mean, he was interested
in topological magic--
like how you can
braid something,
how you can take
a piece of paper--
ignore what's written
on this for now--
how you can take a piece of
paper and cut two slots on it
and tie a knot in
the three things
that result. Mind boggling.
He knew about that
in the '40s, and that
was early days of topology.
He started writing about it.
Ultimately, it led to the
"Scientific American" gig.
He was a freelance journalist,
never had to put on a suit
and tie, didn't have a day
job for most of his life.
And he was a terrific teacher
but from a typewriter--
behind an old '70s
electric typewriter.
His word processing consisted
of Elmer's glue and a scissors.
He would rearrange stuff,
old school, very old school.
And amazingly, apparently,
and his biographer
confirms this-- he never gave
a public talk in his life
anywhere, never
performed in public.
It's astonishing because he
was such an articulate person
privately and in writing.
But he didn't like crowds.
And in fact, somebody
commented recently
that if there were
three people in a room,
he thought that was a multitude,
and he wanted to get out.
He just couldn't handle crowds.
Actually, of course, had he
bitten the bullet-- I mean,
it's not easy to talk to a
public gathering the first time
you do it-- but
like anybody else,
he would have got over his stage
shyness and he would've been
a brilliant expositor had he
given himself that opportunity,
but he didn't.
A little bit here on Twitter.
Got a few Twitters, good.
Well, Martin's on Twitter too.
And what would
Martin Gardner tweet?
[LAUGHTER] Now,
being modest Martin,
he wouldn't tweet about
I wrote 104 books.
He would tweet about
oh, here's a cool thing.
Check this out.
Here's a video,
here's a rope trick,
so that's what he tweets.
And he also treated his entire
memoirs about a year ago,
which took a while--
one tweet a day.
But because he's
such a modest guy,
there's the other side
of the story, which
is he did write 104 books.
He wrote 40 books on puzzles.
He wrote two important
books on physics.
He wrote these
books on philosophy.
We have to also tell that
story, so we balance it
with the Martin Gardner
centennial account,
so both of those I would
encourage you to follow
and tell your friends, please.
We'd like to build up
the following with those.
They tweet once a today and
hopefully will for a long time
to come.
The new website, it went
up about a year ago is
martin-gardner.org.
And there's a lot of
stuff there about him.
So here's the next math problem.
How many cylinders or
pencils can be arranged
so they all touch each other?
Think of fingers.
Could you get three fingers
that they all touch each other?
I think I pulled it off.
Could I get four fingers so
they all touch each other?
Well, that might
not be the solution,
but it is if they join
there and they do.
So OK, I've got a solution,
so we can probably do four.
So can you do five?
OK, can you do six?
That was kind of what
Martin was fishing for when
he asked this question.
He showed a picture
of four tennis balls
in a pyramid or golf
balls and three coins
that mutually touched.
And then he said,
what about cylinders,
like long cylinders or pencils?
And he was hoping for six, and
it turns out six is doable.
And here's the
picture he had in mind
when he posed this question
in "Scientific American."
But, he didn't
publish the picture
until later, and
in the meantime,
people found a solution
that he hadn't anticipated.
And actually, can you do seven?
Can you get seven pencils
that all mutually touch?
Well, that was the
surprise, because somebody
wrote in and said, here's six,
but here's seven, and here's
the picture.
You gotta think
outside the page.
It's coming out at you.
The one in the middle,
there's a seventh one there.
So it's a nice symmetry.
The previous picture
didn't have any asymmetry
to speak of-- well,
it had some, but this
is a nicer symmetric picture for
six and with a third dimension
you've got the seven.
Now those pencils might happen
to be hexagonal, but it's OK.
Circular, cross-section,
hexagonal--
we don't really care.
So seven's possible, but
we are using the fact
that some of those pens
are a finite length
because the ends are touching.
So that-- then you say hmm,
does it matter the pencils all
have the same length?
And I made a very bad
mistake about a year ago,
because I Googled, ad
what I saw horrified me.
I shouldn't have Googled, I
should have used my brain,
because I missed
something important.
So, I'll tell you.
It is possible to do eight.
Thank Google.
Find it yourself.
If the pencils have different
length, you can get eight.
What about if the pencils
are infinitely long, infinite
cylinders?
Have any infinite cylinders
could mutually touch?
Now, you could
certainly get three,
you can probably get four.
Not clear about the five
and the six, not so obvious.
The pictures I've shown
you certainly would not
work if the pencils were
infinitely long, so you
have to re-think, redo things.
Well, that's an
interesting question.
Littlewood actually
asked this question
in a book he published in 1966.
He said can seven
infinite cigarettes
be arranged so that each
touches all the others.
And this was an open problem.
Now Gathering for Gardner
happens every two years.
At the last Gathering
for Gardner,
three guys got up onstage
and very excitedly
announced the solution.
We didn't know the answer
until quite recently.
The paper was actually
previewed in 2013.
But we know as of
2013 that yes, you can
get seven infinite cylinders.
And they did it using
a lot of computer time.
Well, they set up 20
polynomial equations
between the variables.
You know, a cylinder
can be modeled
by x squared plus y
squared equals a constant.
And then you have to be able
to put in the different angles
in three space.
And it's not so linear algebra.
It's several equations
and several variables that
are quadratic but
nonlinear, so you
have to use other
math techniques, maybe
Grosvenor basis.
But it was complicated, and it
took a lot of computer time.
But it found two solutions
quite early in the search.
And the fact that they found
two early in the search
means there are
probably a lot more.
I'm going to show you a
picture of one of them.
It's weird because it's
not remotely symmetric.
Now, you see ends there
but that's irrelevant.
I just couldn't print-- I
couldn't-- the slide wasn't big
enough to contain
the infinite ones.
But rather using the
ends, the touching
has nothing to do with the ends.
And here's the
fascinating thing,
and this just boggles my mind.
This could have been
discovered many, many times
before by a kid with some
straws-- mommy, mommy, mommy.
You know, anybody could have
come up with that, right.
Just cluster them
together in their hand,
and it might have just
happened that they touched.
And yet, it took all the king's
horses and all the king's men,
and a lot of computing time
to figure out it was possible.
So even in three
space, our grasp of it
is not as firm as
you might think.
It's kind of surprising.
Can you do eight?
Is there a mathematical reason
in terms of the algebra why
you can't do 8, 9, 10?
Here's the best result known.
The most you could
ever have would be 24.
Probably not.
I mean it probably would
be lower than that,
but we can't prove that.
What we're absolutely sure
is you can't get 24 or more,
but we don't know if
you can get 19 or 9.
It's crazy, so we--
thank you-- so we
got the five-- we got the
seven, but we're not quite sure.
But a nice quote here
from Dana Richards.
The aha moment--
you think it out,
and you spot it and
use it, you go, aha!
I got it.
And then it raises
new questions.
The cycle of enlightenment
and new questions.
And that's kind of
what Martin engenders
in many generations of people.
Dana Richards is a
bibliographer and biographer.
Now, is there a three by
three magic square of squares?
Euler, famous mathematician
a long time ago,
1770, came up with this
example of a four by four.
So magic square means
that if you add up
all the rows and columns
and diagonals and perhaps
some of the things, you
get the same constant.
But each number in the
square has to be a square,
and that makes it
extra complicated.
So Euler came up with
the solution for a four
by four magic square of
squares, distinct squares.
If you make them all the same,
it's not very interesting.
But he wasn't sure
about three by threes,
and people have been worrying
about this ever since.
And quite recently,
only a decade ago, a guy
called Christian Boyer came
up with five by five and six
by six and seven
by seven examples.
But we still don't know
about three by threes.
Nobody's ever found
one, and nobody's ever
proved it's impossible.
And trust me, they've tried with
numbers up to five gazillion.
So, don't try it with
your-- it's been tested.
If there's a solution, it's
way out there, way out there.
So Martin Gardner
offered a prize for this.
[LAUGHTER] Nobody's
claimed it yet.
And I will speak off the
record on behalf of the family
that if somebody
nails this, I'm fairly
confident that the family's
estate will pony up
the $100, so go for it.
[LAUGHTER] This'll be the most
famous $100 you ever made.
There's a related
question, by the way,
about magic squares where
all the elements are primes.
And there was a breakthrough
in that more recently.
That was hard for a while too.
So here's the best three
by three example anybody's
come up.
Lee Sallows is a British
engineer who lives in Holland,
and he came up with this
example of-- it's almost right,
but it turns out that all
the rows and columns and sums
are 21,000 sums of 9 except
the top left to bottom right
diagonal.
One messes up.
You can also get solutions
where everything adds up right,
but one of the numbers
isn't a perfect square.
So like a little
compromise, you can do it,
but if you want to get them
perfect, we're not quite sure.
Those numbers are
obviously too small.
We know there's no
solutions for numbers
up to a very large number.
OK, here's the last problem
that I'm going to ask you.
Martin Gardner asked this
question in 1990, 25 years ago.
What is the minimal surface
area inside a transparent cube
that will render it opaque?
And this room is not a cube.
It's very long, but just
pretend it's a cube, right.
We're going to squish you
all into a sardine tin,
we're going to stretch
the roof a bit.
Imagine it's a cube.
What system of
interior walls would we
have to erect so if the ceiling
and the floor and the four
walls were all passed, no
light beam could pass through?
We want to block
all light beams.
Well, you could just take
the four walls and the floor
and just duplicate them
with interior walls,
leave the ceiling open, so
five cubic units of everything
one by one by one would work.
You can do better than that.
Maybe you could put a
diagonal wall across here,
a diagonal wall from that
corner to that corner
and an eye to the
roof or the ceiling.
So you can cut down.
It turns out that
any question like
this in three dimensions you
might say well, let me first
go back to the 2D version
and see if I can solve that.
So let's back off
to two dimensions.
So we're taking about a square.
You're in a square, and
you want to stop light rays
from passing across the square.
Well, the two diagonal
walls would work,
and they have a
length of 2 root 2.
Actually that's the
picture on the left.
The picture on the right
here is a little better.
Those two mysterious
points, it almost looks
like a folded up envelope,
are called Steiner points.
And this here has the
property that it's
the most efficient interior
road network, the shortest
interior road network
connecting the four corner
cities or towns.
So that seems like
a better deal,
and it comes out to be
1 plus root 3, which
is 2.73 down from 2.83.
We're making progress or are we?
Well, it turns out
you can do better.
And the one I'm going
to show you now is 2.64.
We've come down a bit, but
it's really surprising.
It's got some symmetry,
but it's also disconnected.
Wouldn't make a very
good road network,
wouldn't recommend
it for driving.
And Ken Brakke who developed the
surface evolver about 25 years
ago came up with this.
And I mean, that to me
looks like a flapping door,
so I say that's-- it leaves the
door open to the possibility
of future improvement.
This is the best we've known.
We just don't know
if we can do better.
This is a square.
We can't even solve
it for a square.
As a matter of
fact, we don't even
know the answer for a triangle.
Think of a Mercedes Benz
symbol, right, Steiner
point in the middle.
That's believed to be
the best for a triangle,
but we don't know.
We're not sure.
So we can't even do
it for a triangle,
an equilateral triangle.
Astonishing.
How hard can it be?
Well, here's the best
solution for the three
by-- for the three-dimensional,
and Ken Brakke
came up with this.
This is like a
tent or something,
and it's highly non-symmetric.
And this was 1991 or so.
And we don't know--
24 years since, we've
made no progress on
this that answers
Martin's original question.
So these questions
he asked, posed--
got people going on research.
He's a very simple one,
one of Martin's favorites.
Can you change the giraffe
into a different giraffe
by moving just one toothpick?
And I promised you I'd
go back to the problem
about the Tuesday, so
let's look at this.
So I'm going to show
you the solution,
and I'm going to
invite everybody
in the room to close their
eyes if they haven't already
solved it.
And I'm hoping to see
lots of closed eyes.
And-- but if you
really want to see it,
I'll just let you
look at it silently.
But you're better off solving it
yourself later so-- by the way,
the problem-- Gary
Foshee spoke it,
but it's due to two
mathematicians with PhDs, Mike
and Tom Starbird, brothers.
So here's the solution.
OK, so close your eyes if
you don't want to see it.
And I'll point to something
here for those of you
who are looking.
This is the key thing here.
When you set the
sample space, there's
something you have to pay
attention and go, oh, uh-huh,
hmm.
Well, in that case, ah, hmm.
And look at the answer
if you're looking.
Right, not the
number you expected.
OK, you can look up now, please.
We've got a minute to go.
Bob Crease who's a philosopher
wrote a great piece
on Martin's centennial in
"Physics World" last year.
And he said Googling
is not the Gardner way.
Yes, look at Google to find
out about the Game of Life,
but don't look up Google to find
out the answer to questions.
And our students do that.
Of course they do that,
and that's a problem,
because if you actually try to
discover something yourself,
you learn a lot more.
Randi sent a great comment last
night that he was once asked,
is Martin Gardner one person or
maybe Isaac Azimov and Murray
Gell-Mann combined?
[LAUGHTER] The man was so
prolific in so many fields
there were people in
a pre-internet era
who aren't quite sure
that the guy was real.
So maybe it's a pseudonym
for a pool of talent.
He was one guy.
So every October, November,
the Celebration of Minds
happen, and they happen because
people make them happen.
And I mean you, your friends,
your family, even your enemies.
So please consider running one.
It can be to do with anything
Martin was interested in.
There's a website
where you can register.
There's a website where
you can get ideas.
We've love to see more
Google-affiliated ones.
There's cards over here
if you didn't get them.
They're Scott Kim's design.
First, across the top, it says
Martin Gardner's Celebration
of Mind-- the same thing
in four different fonts.
But you turn them upside
down, and they say four
completely different things.
The existence of any one of
these is pretty miraculous.
The existence of four
that are kind of similar,
one of them completely
in the other way,
is actually miraculous.
You can pick up copies there.
I'll leave you with this.
Jim Propp said that before
there were search engines,
there was Martin Gardner.
When scientists and math people
wanted to find out stuff,
they often wrote
to Martin Gardner.
He was a hub.
He would take stuff and send
it back to somebody else.
And that's kind of the way
things worked until we moved on
to the modern era.
So I thank Google for providing
this opportunity for me
to come speak to you
today, and thank you
very much for your attention.
[APPLAUSE]
