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All dogs are the same color! (Posted on 2003-05-30) Difficulty: 3 of 5
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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re: My thoughts. | Comment 2 of 23 |
(In reply to My thoughts. by Bryan)

The assumption was that it is true for k=1 (so every group of 1 dog). Then the "proof" proceeds by mathematical induction, that is, if it's true for a given k, it's true for k+1.
  Posted by Charlie on 2003-05-30 10:14:59

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