Welcome back, Future Einsteins.
My name is Yao and today you are going to help my friend Ashley solve this problem.
Now let's get started.
Let's first read the question.
My friend Ashley gets up early in the morning and she needs to wear socks by herself.
Ashley has 6 pairs of socks in her drawer: black, blue, purple, yellow, green,
and pink.
Ashley is too tired and takes the sock up one by one from her drawer with her eyes closed.
Then how many socks shall Ashley take out at least to ensure there is a pair of socks to wear.
Okay. So, let's first draw a graph.
2 black, 2 blue, 2 purple, 2 yellow, 2 green, and 2 pink.
So most people will think....What if she got purple and then got another purple.
The she'll get a pair.
But it says then how many socks shall Ashley take out at least to ensured there is a pair of socks to wear.
So even on her own unluckiest day she'll get a pair of socks.
So let's see.
Let's say she got black.
Second, she got blue.
Third, she got a purple.
Fourth, a yellow.
Fifth, a green
Sixth, a pink and let's see all the seventh try whatever she gets she will get a pair.
Like black, pair up with this.
Blue(with the other blue)
Purple, yellow, green, and pink are all the same.
So it takes 7 turns, for Ashley to get a pair of socks.
So Ashley take seven turns to ensure she gets a pair of socks to wear today.
Even if she's like so unlucky, she will still get a pair of socks.
So this question is easy but today I'm going to introduce you a new principle called the pigeonhole principle.
The pigeonhole principle states that if N plus 1 items put in N containers there at least one of them has more than one item.
And there's also a extension for this principle.
If KN plus 1 items is put in N containers there must be at least one container that has more than K items.
Even though this principle is easy, it is famous throughout the world.
Now that you know this, try the second problem on your own.
Let's read it.
In an equilateral triangle with the side length of 2, select any 5 point arbitrarily.
Can we guarantee that there must be 2 points whose distance is less than or equal to 1?
Okay, solve the problem.
Okay, are you done?
Before we start this question, let's erase it.
Okay.
So it says the equilateral triangle we know it's an equilateral triangle 2, 2, and 2.
Remember in the last question, it gave us
the pigeonhole which was six since six pairs of socks
And here it didn't give
us it so we're gonna make them ourselves.
Midpoint of this line, Midpoint of this line, and Midpoint of this line.
Let's connect them together.
Okay, so there are 1, 2, 3, and 4.
4 pigeonholes and there are five points
since it says that.
Until now, we have four pigeonholes.
Every pigeonhole is an equilateral triangle with side length one.
It says select any five points so it has
four pigeonholes and five points.
According to the principle, there must be
one equilateral triangle with two points.
So let's draw an equilateral triangle.
It could be any of these four and since the side length is one.
That is the longest of the two points connected in this equilateral triangle.
So that means anywhere on this triangle there must be two whose distance is less than or equal to one.
Okay, we solved it.
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See you next time, Future Einsteins
