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This was actually a homework assignment in a math lecture in Germany.

We prove with mathematical induction that all humans have the same gender. So consider a room with $n$ people. For $n=1$ the statement is obviously true.

Now the inductive step: If there are $n+1$ people in the room we ask one arbitrary person to leave the room. So now only $n$ people are left in the room. By the induction hypothesis all these people have the same gender. The person outside now comes back and another person has to leave the room. So again there are $n$ people in the room and all having the same gender. Hence the $n+1$ people all have the same gender.

So were is the flaw :-)

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Consider the possibilities when you started off with two people ... they are ever in the room together when the decisive judgment is made.

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    $\begingroup$ Yes, that means the inductive step is wrong for $n=1$. $\endgroup$ – Michael Lehn Sep 17 '13 at 21:42
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Hence the n+1 people all have the same gender.

Everything up to that sentence was ok. What justifies this deduction? Therein lies the flaw.

If a man and woman are in the same room, and one leaves, yes the room now only contains one gender. The person who left comes back in, and the other leaves. Again, the room only has one gender. Where does the conclusion come from that therefore both people have the same gender?

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As @MarkBennet says, the flaw can be easily seen if you start off with 2 people. Basically, the problem is that you're assuming that if you remove 1 person and have n people left, vs removing another person and having n people left, the two groups of n people have the same color. But this is only true if there's a common person in both groups of n people, and that is not true if you start off with two people. See this link.

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