Originally Posted by

**bobak** Okay were going down this path now......

Statement: All horses are the same colour.

Proof (by induction): For one horse the statement is true. assume the statement is true for any set of k horses, then consider a set of k+1 horses. $\displaystyle ( h_1 , h_2 , ... , h_k , h_{k+1} )$ then take any two distinct k element subsets $\displaystyle ( h_1 , h_2 , ... , h_k ,)$ and $\displaystyle ( h_2 , ... , h_k , h_{k+1} )$ for example. we know that all the horses in the k element subsets are the same colour, so $\displaystyle h_1 = h_2 = ... = h_k$ and $\displaystyle h_2 = ... = h_k = h_{k+1}$. this gives $\displaystyle h_1 = h_2 = ... = h_k = h_{k+1}$. so this prove that for any set of k+1 horse they are the same colour. so by induction all horses are the same colour.

Bobak