Perhaps this has some amusement value. Can these examples be made even more dramatic than they are? ---- Here is a checkers position I created to annoy Jonathan Schaefer, leader of the "chinook" project which solved the game of checkers (forced draw). View in constant-width font. .W.W.W.W W.W.W.w. .w.w.w.. ........ ...b.... ........ ...W.... ........ W=white king, w=white checker, white is moving North. White to move. In this position, white has 8 kings plus 4 checkers, while black has only a single checker. Nevertheless, black soon wins the game. Here's basically the same thing backed up a little to make it a bit more amusing, black to move and win: .W.W.W.W W.W.W.w. .w.w.... ..b...B. .......B ........ ...W.w.. ........ In view of this, should the Chinook team still contend they solved checkers? I took a look at their paper in Science and it looks to me like they really did solve checkers. Their strategy was to use chinook search + heuristic evaluator + perfect endgame tables. They assumed (perhaps falsely) that a heuristic eval > X was a "win" and a heuristic eval < -X was a "loss." They then pseudo-solved the game. Then they increased the value of X and did it again. They only claimed a full proof once X had increased all the way to a genuine win. So this kind of counterexample shows you need to go a pretty long way and cannot cheat by using too-small X... but the chinook team claims not to have cheated (although I think their initial efforts had cheated in this way). Warren D. Smith http://RangeVoting.org