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Old October 29th 14, 08:58 AM posted to rec.games.chess.misc
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Default Endgames on larger boards

On a 10 x 10 board, is bishop and knight and king against king a draw? I would think so, because the win seems to make essential use of the board constraints, but it would be interesting to see it confirmed. How about queen and king against rook and king? Is that always a win no matter how big the board is (ignoring ridiculously small boards)?

I'm not considering the 50 move rule here. If the 50 move rule applies, draws are very prevalent on large boards.

Thank you,

Paul Epstein
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Old November 17th 14, 10:11 AM posted to rec.games.chess.misc
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Default Endgames on larger boards

Op woensdag 29 oktober 2014 09:58:13 UTC+1 schreef Paul:
On a 10 x 10 board, is bishop and knight and king against king a draw? I would think so, because the win seems to make essential use of the board constraints, but it would be interesting to see it confirmed. How about queen and king against rook and king? Is that always a win no matter how big the board is (ignoring ridiculously small boards)?

I'm not considering the 50 move rule here. If the 50 move rule applies, draws are very prevalent on large boards.

Thank you,

Paul Epstein


Your intuition is partly right: short-range pieces often lose their mating potential when the board size is enlarged. For instance, a pair of Kings (one royal, one non-royal) can force checkmate on a bare opponent King on 8x8 in maximally 18 moves. This increases to 23 (9x9), 29 (10x10), 35 (11x11), 42 (12x12), 50 (13x13) and 62 (14x14). For 15x15 the end-game turns into a general draw, with only 0.2% of the black-to-move positions lost. For 14x14 that was still 92.8%, the remaining 7.2% due to immediate capture of one of the white Kings.

The reason is that a successful defense strategy on big boards is to simply run away from the white pieces; perhaps one of them could catch up with you, but unless it is extremely powerful, (like a King-Knight compound ('Centaur')) it would not be able to stop you from increasing your distance to its King, and you could not make two pieces (which together could succesfully restrain it) catch up with it.

For sliders this is of course different, and a Rook has mating potential on any size board. It can always be positioned to obstruct the fleeing King in just a single move. It even has mating potential on an infinite board (a quarter plane).

Now B+N is a tricky case, because you have a mix of a slider and a piece that is faster than a King. So the Knight can always catch up with a fleeing King, and the Bishop can then join in a single move. Together they can cause a lot of obstruction of the bare King. Although the probably cannot really trap it, they might be able to slow down its escape enough for their own King to catch up with the action, and the three of them can then likely shephard the bare King to an edge, and sbsequently a corner. So it cannot be excluded that KBNK would actually be a win on any size board, or even on an infinite one.

The actual longest mates also suggest this would be the case: on 8x8 KBNK takes 33 moves, and this increases to 47 (10x10), 64 (12x12) and 78 (14x14). (Could not build tablebases bigger than that, on my 4GB machine). On 6x6 it takes 22 moves. This does not show a faster-than-linear increase like it does for K+K vs K. It seems that for every enlargement of the board you just need to repeat the same manoeuvre one extra time, taking about 14 moves (which includes the 3 extra moves you might need to move your King and Knight to where the action is).
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