Which endgame mate is faster?

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Sqod

Here's something I've wondered a long time, and I never took the time out to analyze it, so the quick way is to just ask...

If you're in an endgame and you have two pawns and there are no other pawns or pieces on the board other than the kings, is it faster to queen one pawn and mate with K + Q, or faster to queen both pawns and mate with Q + Q? I'm sure it makes a difference how close the pawns are to queening, and maybe where the opponent king is, but assuming the opponent king runs to the center of the board at the time you queen one pawn, is there a rough way to estimate for whether it's worth the time march the second pawn in to queen for a faster mate? The reason I keep running into this question is that when I play against a computer it never resigns so I have to play the game all the way to the bitter end, and some programs take a long time to calculate each move even in ridiculously lost positions like that, so it would be nice to know a good rule of thumb to hasten the mate.

notmtwain

This is one of those questions you can actually answer using endgame tablebases like http://www.shredderchess.com/online-chess/online-databases/endgame-database.html

Put the pawns and kings on the board and try it.

I tried a half a dozen positions and it seemed like the rule of thumb was to try to herd the king to the side with your own king.

Sqod

Nice, thanks. That's a better way to go about it. Otherwise I might have to play a lot of games with each strategy, multiplied by several different initial setups per strategy, in order to develop some crude statistics, which would take a long time, which is why I never got around to doing such analysis.

Sqod

Follow-up:

I used the great endgame database site that notmtwain suggested, and by systematically setting up a set of typical endgame queening positions I was able to get what I believe is a good, reliable rule of thumb: If it takes 5 moves to produce a 2nd queen, then it is equally fast to mate with just the first queen. Or equivalently: If it takes *less* than 5 moves to produce a 2nd queen then it is faster to march the 2nd pawn to queen and then mate with the resulting two queens, otherwise, it's faster just to mate with one queen.

Below are my calcuations, notes, and reasoning. Let me know if anybody sees a flaw anywhere...

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ASSUMPTIONS

()
Assumption: The King of the superior side will be on the 7th or 8th rank alongside at least one of the queening pawns, so as to protect it while queening.
Position used: King of the superior side at a7.
()
Assumption: The King of the inferior side will be near the center, probably while running from the protected pawn that was about to queen.
Position used: King of the inferior side at d5.
()
Assumption: The pawns most likely to queen are those near the edges, since those are the ones that aren't captured in the center early in the game.
Position used: King of the superior side at a7.

CONVENTIONS
White is the superior side.
White is the side to move.
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White K at a7
Black K at d5
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NUMBER OF MOVES UNTIL MATE
--------------------
ONE QUEEN

White Q at b8: 7 moves
White Q at c8: 8 moves
White Q at e8: 7 moves
White Q at f8: 7 moves
White Q at h8: 8 moves

1-Q average = 37/5 = 7.4 moves
--------------------
TWO QUEENS
----------
1st White Q at b8...

2nd White Q at c8: 3 moves
2nd White Q at e8: 2 moves
2nd White Q at f8: 3 moves
2nd White Q at h8: 2 moves

2-Q average with this 1st Q placement = 10/4 = 2.5 moves
----------
1st White Q at c8...

2nd White Q at e8: 2 moves
2nd White Q at f8: 2 moves
2nd White Q at h8: 3 moves

2-Q average with this 1st Q placement = 7/3 = 2.3 moves
----------
1st White Q at e8...

2nd White Q at f8: 3 moves
2nd White Q at h8: 2 moves

2-Q average with this 1st Q placement = 5/2 = 2.5 moves
----------
1st White Q at f8...

2nd White Q at h8: 3 moves

2-Q average with this 1st Q placement = 3/1 = 3.0 moves
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STATISTICS FOR 2 QUEENS

2-Q average with this 1st Q placement = 10/4 = 2.5 moves
2-Q average with this 1st Q placement = 7/3 = 2.3 moves
2-Q average with this 1st Q placement = 5/2 = 2.5 moves
2-Q average with this 1st Q placement = 3/1 = 3.0 moves

2-Q average = (2.5 + 2.3 + 2.5 + 3.0) / 4 = 10.3/4 = 2.6 moves
--------------------------------------------------------------------------------
CONCLUSIONS

1-Q average = 7.4 moves
2-Q average = 2.6 moves

difference = 7.4 - 2.6 = 4.8 = approximately 5 moves

IMPLICATIONS

(1)
   Endgames on an empty board with only one queen take approximately 5 moves longer to mate than equivalent positions with two queens.
(2)
   If it takes more than 5 moves to queen a pawn (including any move where the superior king must move out of the way of the pawn or move to protect a pawn),
then it is faster to mate with king and only one queen.
   This means, in the case of an unobstructed, constantly unthreatened pawn, that the pawn intending to queen must start moving:
(a) From the 3rd rank in order to *match* the speed of mating with only king and one queen.
(b) From the 4th rank or higher in order to *increase* the speed of mating with only king and one queen.

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