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| XGID=-b----E-C---eE---c-e----B-:0:0:1:00:0:0:0:3:10 | |||||
3-point match, Blue rolls first | |||||
In a 3-point match, both sides play perfectly (according to XGR++ evaluation). On White's fourth turn (the eighth move overall), with the cube still in the center, she closes Blue out. How is this possible?
First of all, congratulations to Steve Sax, who submitted a perfect solution! And to Ray Kershaw, who came very close.
SOLUTIONS:
The minimum number of pips White can roll to close her board is 64 (by using all the checkers on her near side plus four from the midpoint). If she does exactly that, she will need a minimum of fourteen roll portions -- three from the 6pt spares, three from the 8pt, and eight crossovers from the midpoint, which means that at most one non-doublet is rolled. If, additionally, White's fifth midpoint checker or a back checker moves (though irrelevant to closing her board), then all four of her rolls must be doublets. Furthermore, if a non-doublet is rolled , then it must be a multiple of 4 (i.e., a roll of 31, 53 or 62).
Some roll sets can be easily eliminated. For example, for 22 62 66 66, 62 44 44 66, 22 44 44 66, and higher even-pip combinations, it is impossible to close the board. White's 1pt, 3pt and 5pt need to be occupied by six checkers, whereas only five checkers (the midpointers) are an even number of pips away.
Even for roll sets with a mixture of odd and even numbers, some are just unaccomodative to filling the board. This is true for sets having three rolls all larger than double 4s (with the exception of 22 55 66 66). For another example, 33 33 66 66 turns the third spare on White's 6pt into a dilly. A little less obviously, for roll sets like 31 66 66 66, 62 55 66 66, and 53 55 66 66, there is no legal way to get two checkers to the 4pt.
Nevertheless, I found 19 roll sets that can close the board. Of these, 12 can be integrated with Blue's moves in such a way that both sides play the checkers perfectly while also correctly keeping the cube centered.
Typically, there are multiple roll orders that work with each roll set, lots of move-pair inversions, and numerous alternatives for Blue's plays -- far too many to itemize. Instead, I've listed just one play sequence per roll set, and for simplicity I've had Blue fan as much as possible (though often several entering moves work just as well). That's not to imply these sequences are trivial to find; it's just the nature of backgammon puzzles that so many transpositions exist.
The 12 qualifying roll sets below are listed in pip-numeric order, each accompanied by one perfectly played sequence (at the ¡V3¡V3 match score) where it is never right to turn the cube (always too good or not good enough) yet White closes out Blue. If you are not fluent in Nactation, you can review the tutorial or put the rolls into XGR++ to verify the best plays.
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SOLUTIONS to Problem 1:
..(1).. 11 33 66 66........61P-66O-42P-33A-65D-11P-65R-66P
..(2).. 11 44 55 66........31P-55P-33D-44A-65R-66P-F-11P
..(3).. 31 33 66 66........61P-66O-42P-33A-52D-31P-65R-66P
..(4).. 31 44 55 66........52S-31P-61P-66P-42S-55P-F-44A.......(see also Ray Kershaw's sequence, albeit imperfect)
..(5).. 22 53 66 66........61P-66O-55P-22A-65R-66P-F-53P
..(6).. 22 33 55 66........62S-33D-53U-22A-1-55J_F-66O.
..(7).. 22 44 55 66........61P-66O-55P-22A-65R-55J_F-44Q
..(8).. 22 55 66 66........61P-66O-55P-22A-65R-55J_F-66O
..(9).. 53 33 55 66........61P-53P-65R-55N-F-66O-44Z-33J
(10).. 53 44 44 66........54S-44P-61U-66P-F-44A-F-53P...........(see also Steve Sax's sequences, with colors reversed)
(11).. 33 33 44 66........52S-33A-31U-66P-F-33H-2-44A
(12).. 44 44 55 66........54S-44P-61U-66P-4-44A-F-55J_
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Seven additional board-closing roll sets are listed below. Each accompanying sequence has the smallest error total I could find (at the ¡V3¡V3 score). There always exists a sequence with only one error, but I've prioritized smallest total error size over fewest errors. (Moreover, in some cases, there are zero-error sequences if White is at GG -- not listed.) Can you find better?
Sequence imperfections are summarized in the right-hand columns (error sizes according to XGR++ evaluation are in red typeface).
(13).. 33 44 55 66........51S-44A-43V-33D_F-66P-F-55O........44A (8/4(2) 6/2*(2)) .008......................................total .008
(14).. 33 44 44 66........43U-44O.31-33I_F-66P-F-44Y............43U (24/21 24/20) .020.........................................total .020
(15).. 33 55 55 66........43U-55P-61P-66O_328-33J-F-55A.....43U (24/21 24/20) .020, 61P (13/7 8/7) .006........total .026
(16).. 11 44 66 66........43Z-11I-62U-66O-F-44O.F-66P..........43Z (24/21 13/9) .002, 11I (6/5(2) 6/4*) .035.......total .037
(17).. 62 33 55 66........32U-62H-55M_33A_1-55D_F-66O.....32U (24/22 24/21) .047.........................................total .047
(18).. 33 44 44 55........41U-44I_31-33D_F-55I_F-44J............41U (24/23 24/20) .049, 44I_ (13/5
...................................................................................................6/2*(2)) .189, 33D_ (13/10(2) 8/5 6/3*) .030.......total .268
(19).. 33 44 66 66.........64Z-66B-33B-33A-2-44J_F-66O_......64Z (24/20 13/7) .089, 33B (bar/22 24/21
...................................................................................................13/10(2)) .040, 33A (8/5* 7/4* 6/3*(2)) 276.........total .405
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Time for a feature film. Here is a sample problem-1 solution from the middle of the pack (the #7 roll set, 22 44 55 66):
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| XGID=-b----EBB---cD---abe---bB-:0:0:1:55:0:0:0:3:10 | |||||
Blue played 61P, White played 66O, Blue rolls 55 | |||||
Blue opened with the standard 61P. In reply, White rolled 66 and (as her back checkers are blocked) played O (Outer, 13/7(2) 8/2(2)).
Blue now rolls 55, and...
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| XGID=-b-B--EBB---cB---abe---bB-:0:0:-1:22:0:0:0:3:10 | |||||
Blue played 55P, White rolls 22 | |||||
Blue played 55P (Point, 13/3(2)), this time blocking White's back checkers from moving deuces. However, White is still able to play her double 2s well...
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| XGID=-b-B--EBB---cB---a-cbb-bB-:0:0:1:65:0:0:0:3:10 | |||||
White played 22A, Blue rolls 65 | |||||
White played 22A (Attack, 7/5(2) 6/4(2)).
Blue rolls 65, which seems like a decent roll...
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| XGID=-b-B--EBB---cC---a-cbb-bA-:0:0:-1:55:0:0:0:3:10 | |||||
Blue played 65R, White rolls 55 | |||||
Blue played 65R (Run, 24/13). He is annoyed at having rolled fewer doublets than White, though he managed to block her back checkers in both instances and now salvages the compensation of having one checker back versus two back. Not to be outdone, White presses onward with a roll of 55...
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| XGID=-b-B--EBB---bC-----bbbbbaA:0:0:-1:44:0:0:0:3:10 | |||||
White played 55J_, Blue fanned, White rolls 44 | |||||
White played 55J_ (Jump 1:2:1, 13/8 8/3(2) 6/1*), and Blue fanned.
White (having suddenly gone from not good enough to double, to too good to double) now has her choice of three rolls (each of which represent a way to solve one of the roll sets #6, #7 and #8) that she can use to close her board. I've selected the roll 44 here for two reasons: (1) The sequence is more colorful if a White back checker has been moved, and (2) To showcase the Nactation doublet-letter Q ("Quarter," where one roll portion is played in each area -- 24/20 is far, 13/9 is down, 9/5 is jump, and 5/1 is inside).
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| XGID=-a-B-aEBB---aC-----bbbbbbA:0:0:1:00:0:0:0:3:10 | |||||
61P-66O-55P-22A-65R-55J_F-44Q .....Final Position | |||||
White played 44Q (24/20 13/1), closing Blue out.
[Footnote regarding problem 1: Ray Kershaw authored Backgammon Funfair -- a book I haven't seen, but it sounds like fun. In it, he published a four-roll close-out sequence (see the link next to #4 above). Unfortunately, for 54S-31P-F-66, P (Point, 13/1*) is not the best move (as George pointed out), not even in a 3-point match. As I did myself a couple times, maybe Ray confused it with the parallel sequence 54S-44P-F-66... (where P is right but the sequence is cooked by cubing before the 66); or maybe he checked 3-ply but not ++; and/or perhaps the cube wasn't a factor in his book sequence. If it was the case that bot information was different at time of publication, I know how Ray must feel. When Paul Weaver and I published my first four-score position in Backgammon Openings (2007), it was fully supported, but sadly I later discovered that a GnuBG version (14.3 devel) used in one of the four rollouts was corrupt. I've since published online a correction (along with several other four score positions, supported by XG rollouts). The point, though, is that at whatever the current time, there is a practical limit to how reliably one can proof.]
In a 3-point match, both sides play perfectly (according to XGR++ evaluation). On White's fourth turn (the eighth move overall), with the cube still in the center, she closes Blue out, yet now BLUE IS TOO GOOD TO DOUBLE! How is this possible?
This puzzle possesses all the challenges of problem 1, plus finding a way for Blue to have become too good to double. As he will be closed out, the only way he can be too good is to have White trapped behind a strong prime. In point of fact, it must be a SIX-point prime.
Here is the solution:
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| XGID=-b---BD-B---cE---c-e--b-B-:0:0:1:33:0:0:0:3:10 | |||||
Blue opened with 31P, White played 55P, Blue rolls 33 | |||||
Blue got his best opening roll of 31 (making his 5pt), and White responded with 55P (making her 3pt). Now Blue rolls 33. What should he do?
According to XGR++ evaluation: In a longer tied match (or for money), Blue should play E (Each, 24/21(2) 6/3(2)), with second best being B (Both, 24/21(2) 13/10(2). Third best, though still typically within .010, is D (Down, 13/7(2)). In a 3-point match, however, D is best, toppling E by .003!
After playing half of his rolls (two of four), Blue is well on the path to making a six-point prime. It is worth noting that he has a faster way of making a full prime, even without rolling 66. If (including 31P, but instead of 33 plus whatever) Blue rolls 22 and 44, he can make a prime from his 9pt to 4pt in a mere three moves. However, that combination would accomplish the coveted feat slightly too fast, as we will see.
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| XGID=-b---BDBB---cC---c-e--b-B-:0:0:-1:44:0:0:0:4:10 | |||||
Blue played 33D, White rolls 44 | |||||
Blue played 33D (Down, 13/7(2)), and now White rolls 44. Had White previously made his 3pt with 53 instead of 55, she would now play 44P (Point, 13/5(2)). In the actual structure, though (for reasons of distribution and to not leave a direct ace-shot), White should play her 44 as shown below.
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| XGID=-b---BDBB---cC---a-c-bbbB-:0:0:1:22:0:0:0:4:10 | |||||
White played 44A, Blue rolls 22 | |||||
White plays 44A (Attack, 8/4(2) 6/2(2)), then Blue rolls 22 and...
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| XGID=-b--BBBBBA--cB---a-c-bbbB-:0:0:-1:31:0:0:0:4:10 | |||||
Blue played 22n, White rolls 31 | |||||
Naturally, Blue played 22n (near, 13/9 6/4(2)), making a five-point prime and starting the sixth point in back. Can we allot White a roll of 11 now? No, because instead of making her 5pt, she would anchor on the 22pt. We can assign her only a roll of 31, which she plays in the obvious way...
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| XGID=-b--BBBBBA--cB-----bbbbbB-:0:0:1:64:0:0:0:4:10 | |||||
White plays 31P, Blue rolls 64 | |||||
White played 31P (Point, 8/5 6/5). Note that her last two moves, 44A and 31P, could have been played in the reverse order. This means that technically problem 2 has two solutions, though one is a trivial transposition of the other.
We have now reached the juncture where it is desirable to revisit my earlier note. If (after opening 31P-55P), instead of -33D-44A-22N-31P, the sequence had continued -22N-31P-44D-44A, Blue would have the same position as diagrammed above but with 13/9 added; that is, he would already have a six-point prime before rolling. The trouble with that is that Blue would now have a proper cube, thereby cooking the sequence.
In the actual position, Blue is not yet good enough to double. Yet he still has one roll that permits him to simultaneously break his anchor (otherwise White can't close her board) and complete his prime. That roll is (of course) 64, and you'll see Blue's indicated play added to the board below.
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| XGID=-b--BBBBBB--cA----AbbbbbA-:0:0:-1:66:0:0:0:4:10 | |||||
Blue played 64S, White rolls 66 | |||||
Blue played 64S (Split, 24/18 13/9). White now rolls 66 and has a choice of two plays. The lesser of two evils is to close her board, as shown below.
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| XGID=-b--BBBBBB--aA-----bbbbbbB:0:0:1:00:0:0:0:3:10 | |||||
31P-55P-33D-44A-22n-31P-64S-66P | |||||
With her roll of 66, White closed her board, reaching the final position. Blue, though on the roof and closed out, is now too good to double.
Returning to Blue's last roll, suppose that (instead of 64) Blue had rolled and played a different 6 -- e.g., 65R (Run, 24/13) -- and White closes her board with the same 66P. In that case, Blue has only a five-point prime and he is NOT too good to double. (Indeed, he is not good enough!)
Nack