The game of 3-Euclid

In this paper we study 3-Euclid, a modification of the game Euclid to three dimensions. In 3-Euclid, a position is a triplet of positive integers, written as (a,b,c). A legal move is to replace the current position with one in which any integer has been reduced by an integral multiple of some other integer. The only restriction on this subtraction is that the result must stay positive. We solve the game for some special cases and prove two theorems which give some properties of 3-Euclid's Sprague-Grundy function. They provide a structural description of all positions of Sprague-Grundy value g with two numbers fixed. We state a theorem which establishes a periodicity in the P positions (i.e., those of Sprague-Grundy value g=0), and extend some results to the misere version.