I'll pay US $200 for a complete analysis of the following impartial game, which (for lack of a better term), I'll call the WILD QUATERNARY GAME. The game is played with a number of heaps of beans of various sizes. There are two players. They take turns making legal moves. The person who makes the last legal move *loses* the game. There are three types of legal moves. Each type of move involves selecting one particular heap, and then removing 1, 2 or 4 beans from that heap, subject to the following additional conditions: 1) A single bean can be removed from any heap, no matter how many beans are in it. 2) Two beans can be removed only if the heap contains exactly two beans. 3) Four beans can be removed from a heap, but only if the heap contains more than 4 beans. In the Winning Ways "octal code" notation, this is the misere game .3102. By a "complete analysis", I mean an explicit mathematical description of the first- or second-player winning positions of the game for arbitrarily-many arbitrarily-sized heaps (ie, 'algorithmic'' solutions that boil down to a recursive search of a game tree don't cut it). For example, if I had changed the problem statement so that the last player to make a legal move *wins* the game, a complete description of the second-player winning positions is All positions with a) An even number of heaps of size (5k+1 or 5k+4) AND b) An even number of heaps of size (5k+2). Thane Plambeck thane@best.com 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm