David Gale:
... the following Nim variant. There are 4 piles, ordinary Nim rules, the only difference being that the game ends when three of the four piles are empty.
Thane Plambeck:
Case II: (Two nonempty heaps, sizes m and n). Everything is an N-position. A wiinning move [is] to take everything in one of the heaps.
Case III: (Three nonempty heaps, sizes r,s,t). Assume r <= s <= t. ... Is it hopeless to give a formula for the Case III nim equivalent?
I looked at Gale's Nim briefly on the weekend. With only three piles, emptying a pile loses quickly, and the game is similar to one in which that's illegal. In fact, GaleNIM position (r,s,t) is P if and only if NIM's (r-1,s-1,t-1) is P. Also, you might enjoy proving that GaleNIM(1,r,s,t) is P iff NIM(r,s,t) is P. -- Don Reble djr@nk.ca