# What is wrong with the following “proof” by mathematical induction that all cats are black?

**Question description**

All Cats Are Black? What is wrong with the following “proof” by mathematical induction that all cats are black? Let P(n) denote the statement “In any group of n cats, if one cat is black, then they are all black.”

Step 1: The base case is clearly true for n=1.

Step 2: Suppose that P(k) is true, and show that P(k+1) is true.

Suppose we have a group of k+1 cats, one of whom is black; call this cat “Tadpole.” Remove some other cat (call it “Sparky”) from the group. We are left with k cats, one of whom (Tadpole) is black, so by the induction hypothesis, all k of these are black. Now put Sparky back in the group and take out Tadpole. We again have a group of k cats, all of whom— except possibly Sparky—are black. Then by the induction hypothesis, Sparky must be black too. So all k+1 cats in the original group are black.

Therefore, by Mathematical induction, P(n) is true for all n.

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