The Answers!
Position One.
Black leads 6-2 to 9. Should he redouble? If he does, should White take?
These positions are very difficult to sort out over the board, as they take us into regions where we are not used to calculating winning chances. For money this is such a monster pass that we don't need to figure White's chances, as they fall so far short of the usual take point. At this score though, a Black redouble puts the match on the line. Let's do our risk/gain analysis.
For Black, ND/WIN is Crawford, needs 7 or 91%.
D/WIN wins the match, 100%
ND/LOSE is needs 3, needs 5 or 65%
D/Lose loses the match, 0%
If Black doubles he risks 65 to gain 9, R/R+G = 65/74 = 88%.
Black's doubling window opens at 88%. Does he have that? It's easier to count White's winners than Black's. White wins when Black fails to throw 2-2 or better (31/36) and then rolls 4-4 or better (3/36). That's 3/36 x 31/36 = 93/1296 or about 7.2%. There's also a sequence when Black rolls 2-1 twice and White bears off in one or two rolls which adds about 0.1%, so we'll give White 7.3% wins and Black 92.7%. The double is clear then and as White can preserve 9% of ME by passing, she should drop. Double/Pass.
6 out of 8 contestants got this right.
Position 2.
Money game, Black to play 3-1.
By the slimmest of margins, 12/9, 11/10 is correct. Dave Kettler and Julia both explained this one very nicely for me, pointing out that 11/7 or 12/8 both waste pips after a specific roll next turn. After 11/7 you waste a pip with 3-2. After 12/8 you waste a pip with a 3-1.
4 out of 8 got this right.
Position 3.
Double match point, Black to play 1-1.
Very tough. A 2592 game rollout plumps for 10/9, 7/6, 3/1*. The answer in 1990 was 10/8, 5/4, 2/1* but the bot places that third 0.015ppg behind. The second place play only 0.004 behind is 7/6, 5/4, 3/1*. Perhaps the biggest surprise for me was that 10/7, 2/1* was only sixth best although still quite close. I assume that this is because if White enters with a 6-1, Black will actually do better to have two men sent back! Also, if White dances, Black doesn't really want his next 6 to be played to the ace point, so plays that leave the outfield spare further back do well. Black really must try to pick up all three checkers if he possibly can.
Nobody got this right, with the majority wanting to make the bar.
Position 4.
7 point match, White leads 5-2, cube action?
Double and take for money, but here no. Cube actions that take our opponent exactly to the winning post are usually suspect and this is no double/take.
Black no double/win, trails 4-away, 2-away. 33%
Black double/win, trails 3-away, 2-away. 40%
Black no double/lose, trails 5-away/Crawford. 16%
Black double/lose, loses match. 0%.
Black is risking 16% to gain 7%, R/R+G = 16/23 = 69.5%
Black needs 69.5% minimum to cube and with only 23 winners and 13 losers he only has just under 64%.
4 out of 8 got this right.
Position 5.
Money game, Black to play 1-1.
It's 2/off(2)! In 1990 they got this one wrong and a 5184 game rollout on XG proves it . Basically the safe play wins almost 100% with 99.31 gammons and 17.08 backgammons. Taking 2 off still wins 97% and although the gammons drop to 62%, backgammons jump to 53%. The difference is only 0.007 of a point and I don't think that there is any way to calculate this over the board, but you may like to consider that a considerable part of White's equity after Black takes off 2 checkers resides in White's ability to use the cube correctly. XG manages this very nicely but how confident are you that you know when to redouble against a man who has born off 14 checkers? It's pretty tough. In practice this should make the 2/off play clearly right.
5 out of 8 got this right.
Tomorrow (really!) I'll post the answers for positions 6 through 10.
After these five, Timothy Chow leads with an excellent 4/5.
Enjoy the game!
8 comments:
on problem 2, you are trying to leave yourself with no misses after your next roll or as few as possible. so the goal is to leave no more than one man out, and to have that man no further away than the 8 point. 12/9, 11/10 and 11/7 are equivalent for all rolls except 3/2. with that roll, 11/9 wastes a pip with the crossover, but the correct answer doesn't.
Any chance you can restore the deleted comments now that the answers are published? I'm particularly interested in seeing Tim's answers since his reasoning is always enlightening, even when he gets the wrong answer.
Thanks Dave, I've edited the post.
Sorry Clem, I'm afraid that I have lost the original comments. I hope that mine will be of some help.
Ok. I saved mine and have reposted them in case anyone cares to read how I mis-evaluated the various positions.
Thanks for your enthusiasm, ah_clem, but unfortunately, I did not save my comments. I confess that I am rather startled that Paul would delete non-spam comments without saving a copy. I hope that in the future, Paul will save copies of comments that he deletes. I put quite a bit of effort into my comments and it's a bit disappointing that they're now lost.
Anyway, Positions 1 and 4 are just routine calculations if you've memorized your MET out far enough and have plenty of time. I know my MET out to 7-away/7-away, which is more than enough for these two problems.
Position 2 I recognized from one of Kleinman's books and remembered the answer. Again you can calculate this one exactly because you just have to consider a handful of bad sequences for Black that don't get him off the gammon.
In Position 3, I played 10/7 2/1*. I am not surprised that some other plays may be better, but if the top few plays are close, then I wonder whether 2592 games is enough, and whether the answer would change if one did a 4-ply rollout.
In Position 5, I ultimately "cheated" by playing the QF card but I did make an attempt to estimate the wins, gammons, and backgammons after each play. I think my estimates were pretty good, but it took me a long time and I'm not inclined to reconstruct the calculations now.
Sorry Timothy, the deleted comments were deleted by accident. I had intended to restore them of course.
Michael Depreli got interested in Position 3 and rolled it out using XGR+ for the checker-play decisions. He posted the results on the BGOnline website, and found that 10/8 7/6 2/1* was the top play.
By the way, has Stiefnu still not claimed his prize? I'm still curious as to what it was.
Thank you Timothy. I note that MD didn't credit this blog or the original authors of the position.
Posting rollout results kills discussion. I would rather listen to three decent players discuss a position than look at rollout results. Backgammon isn't about finding out what the answer is, it's about finding out why it is the answer, because if we know that we can apply it over the board.
Stiefnu's prize was two small backgammon booklets, both very old and cute. I should have photographed them before they went!
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