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At any gathering of n people must have the same hair colour.

Step 1, if there is one person at the gathering then everyone at the gathering has the same hair colour.

Step 2, Assume that k gathering of people must have the same hair colour.

Step 3, Some people are gathered. One is sent out of the room, the remaining k people must have the same hair colour. Now bring that person back and send someone else out. You get another k group of people, all who must have the same hair colour.

Therefore all must have the same hair colour.

Meggs 1102
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  • In an induction you need to prove that $P(n)\implies P(n+1)$ for all $n$. – Angina Seng Nov 28 '19 at 07:33
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    Well. I guess in step 3. the problem is that in the second subgroup of $k$ people, all may well have the same hair color, but there's no guarantee that it is the same color of the previous subgroup. Take two people with two different hair color, e.g. The experiment would work, but you could not draw that conclusion. – dfnu Nov 28 '19 at 07:51

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Step three is missing the proof that if all the groups formed by sending someone one are homogeneous, then the whole group is homogeneous.

This breaks with two people because the person you send out need not have the same color as the other.

In fact this "proof" is no proof at all (independently of the fact that it is wrong), because step 3 is just a restatement of the theorem, with the only justification "therefore", but no argument.