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Can you prove in two different ways that at this moment, some two persons on Earth have exactly the same number of hairs on the head?

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  • A lot of people are bald.

  • There are more people than there are hairs on a typical head (by more than the number of bald people).

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    $\begingroup$ You may want to back up the last claim with some numbers. It appears it mostly ranges anywhere from 90000 to 150000, with variations in hair colors. $\endgroup$ – Tim Couwelier Oct 7 '14 at 9:00
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    $\begingroup$ @TimCouwelier: Well, it’s pretty well known it’s about in the hundreds of thousands and that people are in the billions. $\endgroup$ – Ry- Oct 7 '14 at 15:13
  • $\begingroup$ Bald people generally still have hairs on their head (e.g. many "totally bald" men can still grow a beard). That point should really be narrowed to talk about alopecia totalis and alopecia universalis. $\endgroup$ – Peter Taylor Oct 8 '14 at 9:07
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    $\begingroup$ I'm pretty sure at least two people in the world have alopecia totalis at any given time, given that 1 in about 200,000 people have it. $\endgroup$ – Joe Z. Oct 11 '14 at 0:09
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pigeon-hole principle (http://en.wikipedia.org/wiki/Pigeonhole_principle) If we can assume that the world's population of non-bald people are more than the average hair capacity of an individual, we have a 100% probability of atleast two people with exactly the same number of hairs.

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Let's involve some maths in this question. I'll make some assumptions in order to simplify.

Let's say humans have a perfect spheric head, with a surface of $S=4 \pi r^2$ and a radius of $20\text{ cm}$ (which is way bigger than the actual human head radius). This gives a surface of roughly $0.5$ square metres.

Let's say it is full hair covered, and that the density of hairs by square cm is $500$ (way bigger than real hair density) This gives a meter squared density of $5\,000\,000$ hairs which, multiplied by the surface previously calculated gives a total number of $2,500,000$ hairs as our max number of hairs a human could have in its head.

But the number of humans in the earth is actually bigger than $2,500,000$ which means that, when you have found (if you can) a series of humans whose number of hairs goes from $0$ to $2,500,000$ the number of hairs the next human you chose has is going to be between $0$ and the max we calculated.

This means you have found two people with the same number of hairs.

However, this is just an aprox. calculation but as the number of hairs we've got is way bigger than the real number it means that the initial statement or question is also true.

Q: Which would be the smallest population needed to find for sure two persons with the same number of hairs?

(This means getting a more accurate calculation of the max number of hairs.)

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  • $\begingroup$ Lets increase the probability with a little rectification. The head being hemispherical, surface Area =2πr2. $\endgroup$ – thepace Oct 19 '14 at 6:01

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