Prove that if the Nim sum is not zero, then one of the piles is bigger than the Nim sum of the all the other piles.
I've already proven that if the Nim sum of the piles is zero, then any one move will leave a nonzero Nim sum, and if there is a pile with more stones than the Nim sum of all the other piles, then there is a move that makes the Nim sum equal to zero. However, I have problem coming up with the proof for this one and would appreciate some insight into it.
If the Nim sum is not zero, then there is a most significant binary digit that is not zero. At least one of the piles must have that binary digit not zero as well. Now try to prove that this pile is bigger than the Nim sum of all the other piles.