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| XGID=-b----E-C---eE---c-e----B-:0:0:1:00:0:0:3:0:10 | |||||
The worst move is best. What is the opening roll? | |||||
Blue and White agree to play a money game, where the reply roll is automatically the same as the opening roll. (After one play each, the rules revert to normal.)
With the benefit of that foreknowledge, the winner of the opening roll correctly plays what would normally be the worst legal move! (according to XGR++ evaluation).
What is the roll?
Congratulations to Bob Koca, Brian Lonergan and Joe Russell for solving the "Worst is Best" mystery!
[Sorry for my delay in posting this follow-up. I informed three additional respondents their answers (54, 54, 41) were incorrect, and I gave them an extra week to find the correct answer.]
It is possible to solve by brute force (testing to see if the "worst play" for each roll can be improved on). However, there are a couple of useful observations that make the problem more interesting:
(1) What do the worst play for each of the 15 opening rolls have in common? The answer is simple: They all play two checkers off the 8pt (uniquely breaking a point and leaving at least two direct shots, and usually three, to the opponent's back checkers).
[Someone (Paul Weaver?) once informed me he knew of two players that played a money prop. One "spotted" the other an opening 43, with the proviso that he had to play both checkers off the 8pt. (Has a more lopsided prop ever been played?) I'm guessing that this occurred in the mid/late 1970s, after Backgammon in Blood was published, and some players were experimenting with how far they could push aggressive early game slots.]
(2) The 24pt and the opponent's midpoint are 12 pips apart. Cutting 12 in half gives you 6, and cutting 6 in half gives you 3. If you move either of these numbers off of either point, the opponent's mirrored reply roll can automatically hit on a desirable point (there is nowhere safe to hide), and that hit overrides everything else.
The horror of 63! Given that it is too costly to move off of either the 24pt or the midpoint, that leaves the 8pt and 6pt. (No roll other than 63, with the identical reply roll pending, causes this apparently self-destructive restriction.) Moreover, it is illegal to play an opening 6 from the 6pt. The field is thereby narrowed down to just two plays, which I've diagrammed below (side by side, if your window is wide):
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| XGID=-bA--AE-A---cE-a-cae----B-:0:0:1:00:0:0:3:0:10 | |||||
Blue made his worst play | |||||
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| XGID=-bAA--D-B---cE-a-cae----B-:0:0:1:00:0:0:3:0:10 | |||||
Blue made his second-worst play | |||||
According to XGR++, Blue's 63 play depicted in the first (left-hand) diagram is an error of .383, and his play in the second diagram is an error of .290. These are easily the two worst legal 63 plays in normal backgammon. Between these two choices, the worst-play-by-far of J (Jump, 8/5 8/2) leads to a better result for Blue than his second-worst play of A (Attack, 8/2 6/3). White, with her 63, should reply with D (Down, 13/10 13/7), which gives Blue –.021 and –.030, respectively. When you know your opponent will not hit, there is a bit more advantage slotting the 5pt rather than the 3pt, despite the inferior distribution.
Any other 63 play for Blue is easily better, but NOT if you know your opponent is also rolling 63. Foreknowledge is mammoth.
I once played in a pro-am in the Bahamas. My (amateur) partner was happy to go along with my first few plays and I felt relaxed that at least from that perspective it would be a low-stress match. However, not far into the first game, I made a (correct) play that worked out poorly solely because of our opponents' roll. My partner made sure that I knew it. When every few rolls that situation recurred, my partner got increasingly upset that my plays weren't working, and he felt it necessary to repeatedly explain where I went wrong in hindsight. Needless to say, I never played with him again.
The next time someone tries to play Monday morning quarterback with you, explain to them that opening 63J (Blue's nearly quadruple-whopper in the left-hand position) by one-ply rear-view comparison can turn the normally best play of 63S into a blunder. Such examples in backgammon are very common, though; I assure you.
Nack