The puzzle was basically this:
In a money game, from the opening position, Blue turned the cube some time prior to stacking ALL fifteen of his checkers on the same point. Subsequently, White became TOO GOOD to redouble (from 2 to 4), never having hit a checker the entire game! Moreover, both players played the cube and checkers perfectly, according to XGR++ evaluation.
..(A) How is this possible? (Just the general idea, exact sequence not required.)
..(B) How can it be achieved in the fewest rolls?
SOLUTION
(A) The general idea is this: White is hit, so that he gets (at least) a third checker back. He ends up with a high anchor in Blue's board and a checker on the 24pt (it can be on the roof for a while, but must enter). Bearing in, Blue is blocked by the anchor, and his rolls are such that he must stack all fifteen checkers on his 6pt.
At that juncture, White has a closed board (for best gammon chances) and rolls big doubles so that his anchor can run around into his own outfield. This gives him sufficiently powerful racing chances in the event that (from the 24pt) he misses the upcoming double shot, in order to justify being (suddenly) too good to redouble.
(B) The shortest sequence in which this can be achieved is 16 rolls (i.e., 8 for each player). Here is an abbreviated illustration of the solution:
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| XGID=-----bEBC---dCa--cBe------:0:0:1:54:0:0:1:0:10 | |||||
62S-66B-54@-54 | |||||
White opened with 62S (24/18 13/11), Blue got his best roll with 66B (24/18(2) 13/7*(2)), and White anchored with 54@ (bar/20 24/20), to reach the above position.
I thought there might be a way to shorten the sequence by one more roll, by having Blue open with 65R (24/13), then 66Z (24/18 13/7(3)) on his next turn, following the theme of this Big Stack solution. However, this path is foiled because -- given perfect-play constraints -- either (1) Blue cannot find a way to also send a third white checker back, or (2) at some point he will hit on the 1pt.
In the above position, Blue rolls 54 and will run his bar point checkers, hitting in the process.
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| XGID=-a---bEBC---dDA--bacb-----:0:0:1:62:0:0:1:0:10 | |||||
62S-66B-54@-54R-11Y_62 | |||||
From the previous diagram, Blue broke his anchor and ran with 54 (18/14* 18/13), White entered with 11Y_ (bar/24 8/7 6/5(2)), and Blue rolls 62.
From the opening position, there is a transposition, which is 63S-66B-54@-53R-11Y_63, the difference being that Blue's 14pt checker is one pip back on the 15pt, but to make up for it he now rolls an extra pip (63 instead of 62). In either case, Blue will safety his blot and stack it on the 6pt.
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| XGID=-a---bFBC---dD---aabbb----:0:0:1:66:0:0:1:0:10 | |||||
62S-66B-54@-54R-11Y_62T-42P-66 | |||||
From the previous diagram, Blue stacked on his 6pt with 62, White made his 4pt with 42, and Blue now rolls 66.
According to XGR++ evaluation (which I stated in advance would be the arbiter), D (13/7(4)) edges out P (13/1*(2)) by a mere .002. It was difficult to find a combination of two lead-up rolls for White (noting that the first roll must contain an ace so that he can enter on the 24pt) whereby Blue's best play with with 66 is NOT to point on the 1pt. (I'm not sure my search was exhaustive, but I think 11 plus 42 was the only workable combination.)
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| XGID=-a---bFFC---b----bbbbb----:0:0:1:00:0:0:1:0:10 | |||||
| 62S-66B-54@-54R-11Y_62T-42P-66D-65D-C Blue on roll. Cube Action? | |||||
From the previous diagram, Blue played 66D (13/7(4)), White played 65D (13/8 13/7), and Blue can now offer a smallish initial double.
At the time that White rolled 65, the only three-roll combination with which White could close his board was 65 55 66. Moreover, he had to have rolled the 65 first, because had he rolled bigger, Blue would not have enough of an advantage to squeeze in his cube.
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| XGID=-a---bKD-----------bbbbbb-:1:-1:1:11:0:0:1:0:10 | |||||
| 62S-66B-54@-54R-11Y_62T-42P-66D-65D-C-11O.55J-11O-66O-11 | |||||
After Blue doubled, he rolled three sets of double 1s. Meanwhile, White closed his board with 55 and 66, in either order. (This optional roll inversion, and the opening 63S-parallel, were the only possible transpositions, I believe, to stay within 16 rolls. This whole sequence has threaded the needle.)
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| XGID=-aAA--M----------b-bbbbbb-:1:-1:-1:00:0:0:1:0:10 | |||||
| 62S-66B-54@-54R-11Y_62T-42P-66D-65D-C-11O.55J-11O-66O-11-66P-43-R? White on roll. Re-cube Action? | |||||
After Blue played his third double 1s (stacking his last four checkers on the 6pt), White sprung his (no longer needed) anchor with 66P (20/8(2)), and Blue rolled his worst: 43 (forced).
What is the correct re-cube action?
According to XGR++...
- White could never correctly redouble until he got a shot, although a single shot in a position like this (with no checker borne off) would have been enough to warrant a redouble and in some cases a cash.
- White needed double 6s for 24 pips on the previous roll. A mere 20 pips from double 5s (or even 22 pips) failed to give him a big enough advantage in the race, combined with his pending double shot, to justify playing on.
- After Blue's 43, White is TOO GOOD to redouble (11.4% gammons, and a 1-pip-plus-distribution race lead with the non-hitting numbers averaging 10.25 pips). Note, though, that he should stick around once even if he rolls double 3s.
- If Blue had rolled any number other than 43, White would NOT have been too good to redouble. For example, if Blue had double-blotted instead with 52, 53 or 54 (hitting), White would cash; because if he missed, he'd be unable or less able to stick around for subsequent shot(s), and his race would be adversely affected.
Nack