Friday, March 18, 2005

Drawing tendencies in suicide checkers

I (currently) believe that suicide checkers is a draw. In regular checkers, typically endgame positions with equal material and nothing dramatic going on are just draws. Endgame positions with a (numerical) material advantage are normally a win. In suicide checkers, some things are very different. For example, 1 King vs 1 King is never a draw. Depending on who has the move, it is a win or a loss. This kind of observation would make you think that suicide checkers should be either a win or a loss as a whole too - for after all, checkers is a convergent game, i.e. it converges towards endgames because of the fact that you must capture when you can. Therefore, looking at some endgames gives you an idea of what might happen after the middlegame dust has settled. 2 kings vs 1 king is normally won for the side with 2 kings, here's a short example:

[Result "0-1"]
[Setup "1"]
[FEN "W:WK5,K1:BK32."]
1. 1-6 32-28 2. 6-10 28-32 3. 5-9 32-28 4. 9-14 28-32 5. 10-15 32-28 6. 14-18 28-32 7. 15-19 32-28 8. 19-24 28x19 9. 18-23 19x26 0-1

However, if the single king happens to be close enough to the 2 kings, it can sometimes win:

[Result "1-0"]
[Setup "1"]
[FEN "B:WK21,K1:BK18."]
1. 18-22 21-25 2. 22x29 1-6 3. 29-25 6-10 4. 25-22 10-6 5. 22-18 6-2 6. 18-15 2-7 7. 15-11 7x16 1-0

However, all these endings are wins and losses too (This statement is wrong, as George Miller points out in his comment below - there are also drawn positions). Once we get to 2-2 kings though, we start to see some draws; here's an example:

B:WK23,K22:BK11,K8. This position is a draw. And this is where things get really interesting: You can add more or less as much material for the white side to this position without changing the result. Like this: B:WK23,K22,K17:BK11,K8. Draw. Or like this B:WK26,K23,K22,K17:BK11,K8. Still draw - and you could continue! Why is this? If the white kings approach the red kings, the red kings can sacrifice themselves and win. Like this:

[Result "1-0"]
[Setup "1"]
[FEN "B:WK26,K23,K22,K17:BK11,K8"]
1. 8-12 17-14 2. 12-8 26-31 3. 8-12 31-27 4. 12-8 27-24 5. 8-12 22-17 6. 12-8 17-13 7. 8-12 13-9 8. 12-8 9-6 9. 8-12 {the white kings can't go any nearer, else this happens:} 6-2 10. 11-7 2x11 11. 12-16 11x20 1-0

The same is true vice versa. Neither side can make progress, and the game is drawn, despite a clear material imbalance. This means that unlike regular checkers, a large material advantage is sometimes not winning. Here's another example with a king and a man holding the draw:

W:W17,K11,K9,7:BK27,19. Draw.

The implications are rather clear: material is not as important in suicide checkers as in regular checkers, and the game has a drawing tendency because in endgames, often neither side can approach the other without losing instantly.


George Miller said...

I can draw one king vs. 2 kings if you allow me to set the board:) Bet your database can too! Put the single king on a central square - say 18 and the other kings on 1 & 3. Double king side to move.

Martin Fierz said...

George, you are right. Of course, and as usual :-)

Let me rephrase my original statement: with only 1 king left for the weak side, things depend very much on the exact board position. However, with 2 pieces left for the weak side, there are some defensive setups which cannot be cracked no matter how much material the stronger side has, or how the pieces are distributed exactly. Is this better?