Explain what is wrong with the following "proof" by induction of the statement (for n 1) S(n): "Any collection of n rubber ducks consists of rubber ducks of the same colour." "Proof. Clearly S(1) is true. Now suppose that S(n) is true for some n 1. Consider a collection M of n+ 1 rubber ducks. Take one rubber duck, call it x. The induction hypothesis applied to M r {x} shows that the remaining n rubber ducks have all the same colour, say c. Now return x to the set and remove a different rubber duck y. Again, the remaining rubber ducks M r{y}, one of which is x, must all have the same colour c. Therefore, all n + 1 rubber ducks have the same colour c and the result is proved by the principle of mathematical induction."