# All Cats Are Black? What is wrong with the following“proof

All Cats Are Black? What is wrong with the following“proof” by mathematical induction that all cats are black? LetP(n) denote the statement “In any group of n cats, if one catis 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 ofwhom is black; call this cat “Tadpole.” Remove someother cat (call it “Sparky”) from the group. We areleft with k cats, one of whom (Tadpole) is black, soby 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.