Why is this Induction Proof Wrong? 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.

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.

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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.