The Endgames KRK and KQK
This page will give some results obtained by computer about the chess endgames KRK and KQK, both with normal and abnormal rules. The results for normal rules are already known.
Here are some well-known results for the number of moves in endgames, with best play on both sides:
KQK 10
KRK 16
KBBK 19
KBNK 33
For example, the 2nd line means that White, to play, with a King and a Rook can checkmate a lone Black King in 16 moves.
Next we consider increasing sizes of board, starting from 3*3. For KRK the numbers of moves are:
3, 7, 10, 12, 14, 16, ....
This sequence has been continued to size 54*54 by Vaclav Kotesovec, who conjectures that for large n, the number of moves is 7*n/3 + O(1), see this link: KRK
For the endgame KQK, the corresponding sequence starts:
1, 4, 6, 7, 8, 10, ...
This sequence has been continued to size 40*40 by the same author, who conjectures that the number of moves is now 3*n/2 + O(1), see this link : KQK
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On the rest of this page, I give results I have obtained for endgames KRK and KQK with modified rules. When specifying positions, we give the pieces in the order: White King, White Rook/Queen, Black King. For KRK a position requiring 16 moves is (a1,b2,c3). For KQK a position requiring 10 moves is (a1,b2,e6).
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"One Move" Rooks and Queens
From now on we will stick to the 8*8 board and assume White is to play. We will change the rules of chess so as to produce some longer checkmates. An (x) Rook/Queen can only move x squares but can still capture, and therefore give check, at all distances. As usual, an (x) Rook moves horizontally or vertically, while an (x) Queen can also move diagonally. A (1) Queen can easily give checkmate from any position, but now requires 14 moves. A (2) Queen can only reach 16 squares, but checkmate is still quite easy, requiring at most 16 moves. In the table below, these are the only cases where checkmate can be forced from any position, In all other cases, marked(*), we give the largest number of moves which may be required when checkmate can be forced.
x KRK KQK
1 26(*) 14
2 45(*) 16
3 21(*) 22(*)
4 5(*) 20(*)
5 3(*) 15(*)
6 3(*) 17(*)
7 3(*) 4(*)
We add some comments on two cases of interest:
The (1) Rook
Checkmate is easy if the Rook is close to the White King, or can get close. If far away, it can sometimes be captured, e.g. (a1,h8,c3). The longest checkmate is 26 moves from the positions (a1,h2,b3) or (a1,g7,h8).
The (2) Rook
This is more difficult and interesting. The following problems arise:
i. If the White Rook and King are far apart, Black can prevent them meeting, e.g. from position (a1,g7,e5) - or can even capture the Rook, e.g. from position (a1,g7,f6).
ii. The Black King is safe if he can get to the "Bad Corner" - the one which the Rook cannot attack. Thus there is no checkmate from position (c3,b2,a1).
From the position (a1,b2,c3) checkmate can be forced in 35 moves. The longest checkmate is 45 moves from the positions (a1,g7,c6) or (d1,b7,a5). For the human player, it is an interesting puzzle to force checkmate from these positions, disregarding the number of moves.
Here is a shorter but instructive puzzle. From the position (g3,f4,g5), White, to play, can checkmate on his 9th move. But if Black is to play White will require 18 moves.
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"Two Move" Rooks and Queens
We go on to consider (x,y) Rooks and Queens, which can move either x or y squares, 1<=x<y<=7. Two interesting cases are (2,7) and (3,7).
Moves of the (2,7) Rook and Queen
A (2,7) Rook can get from any square to any other but may take a long time. The longest journey is 14 moves, e.g. to go from b2 to g7 - the Rook will make 7 horizontal moves and 7 vertical moves and land on every rank and file. A (2,7) Queen needs only 7 moves to go from b2 to g7, always staying on the long diagonal a1/h8.
Moves of the (3,7) Rook and Queen
With these pieces, it is convenient to divide the ranks into two kinds - ranks 3 and 6 are "bad" and the others "good". The pieces can move amongst the good ranks or bad ranks but there is no connection between the two kinds. Similarly with the files. This means that the board is divided into 4 regions defined by the kind of rank and file: bad/bad (4 squares: c3,c6,f3,f6), bad/good and good/bad (12 squares each) and good/good (36 squares). The longest journey for the Rook is in the good/good region - it takes 10 moves to go from b2 to g7. The Queen takes only 5 moves, again staying on the long diagonal, and has other 5 move journeys in the bad/good and good/bad regions.
In the following cases, apart from the case (3,7) just discussed, there is a single region, i.e. the Rook/Queen can reach all the squares. Except in the cases marked(*), checkmate can always be forced from any position. In the cases marked (*) we give the longest forced checkmate.
x,y KRK KQK
2,3 23 11
3,4 24 12
2,5 27 12
3,5 29 13
4,5 34 16
2,7 30(*) 16
3,7 37(*) 16(*)
The (2,7) Rook
As with the (2) Rook, the (2,7) Rook can sometimes be captured, e.g. from the position (a1,g7,f6). There is no longer a problem with the position (c3,b2,a1), checkmate taking only 6 moves. From position (a1,b2,c3), checkmate takes 27 moves. The longest forced checkmate is 30 moves, e.g. from the position (a1,e5,c3).
The (3,7) Rook.
As we said, the board has four regions:
i. If the Rook starts in the bad/bad region, there is no chance of checkmate - there are no checkmate positions with a Rook on these squares.
ii. If the Rook starts in the bad/good or good/bad regions, checkmate can sometimes be forced, sometimes not. For example, checkmate can be forced in 7 moves from the position (b5,c1,b7) but cannot be forced from the position (b4,c1,b6). The longest checkmate is 19 moves - one such position is (d4,c1,b8).
iii. If the Rook starts in the good/good region, checkmate can always be forced, There are 3 different positions requiring 37 moves: (a1,b2,c3/d3/d4).
The (3,7) Queen
i. Again in the bad/bad region there is no checkmate, since there are no checkmate positions with the Queen on these 4 squares.
ii. In the bad/good region, checkmate can always be forced. The longest checkmate is 16 moves - there is essentially just one position requiring this number: (a1,g3,d5). Similarly for the good/bad region..
iii. In the good/good region checkmate is again always possible. The longest checkmate is 13 moves, e.g. from position (a1,a4,d5).
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According to the above, in the case of KRK, the longest checkmate is 45 moves, obtained when the rook must always move 2 squares. I conjecture that this is the longest obtainable when the move of the Rook is restricted but it still gives check at all distances. How fast does this number increase with the size of the board? (In normal chess, it increases linearly as was stated at the start).
Perhaps longer checkmates can be obtained by restricting the distances at which check is allowed, as well as the moves. If so, how fast does this number increase with the size of the board.?
Of course, similar problems are possible with KQK.
End.
(last modified 27th May 2024).