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 All dogs are the same color! (Posted on 2003-05-30)
Find the mistake in the following proof that all dogs are the same color (if there's any):

Let's use induction. Consider groups of 1 dog. All dogs of every group are the same color, of course. So we now that it's true for 1. Suppose it's true for groups of k dogs, i.e. every group of k dogs are the same color. Then let's consider any group A of k+1 dogs. Consider a subgroup of A containing k dogs. Let's call x the dog in A but not in the subgroup. Then by induction, all dogs in the subgroup are the same color. Now consider a subroup of A of k dogs, with x in the subgroup. All dogs except for x are the same color. Then, since every group of k dogs are the same color (by induction), all dogs in A are the same color. So x and every dog are the same color.

 See The Solution Submitted by Fernando Rating: 3.5000 (10 votes)

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 Error! Comment 23 of 23 |
When k=1, the proof is wrong. If we take k+1=2 dogs, then the subgroups of 1 dog are disjoint. Therefore, they can be of different colors.

 Posted by Math Man on 2011-10-13 20:59:15

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