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With perfect play, reach this position in 8 rolls | |||||
The above position is from Advanced Backgammon #155. When Igor posted it recently, I thought I'd see if it was convertible it into a retro-puzzle. My original post is here, but I'll also repeat the objective below:
OBJECTIVE: Starting with the opening position, reach the above position in a total of eight moves, each player having played four rolls. Either player may start. All checker plays and cube actions have been "perfect" (zero error rate) according to all of XGR, XGR+ and XGR++.
SOLUTION NOTES: White has played thirteen crossovers plus she took two spares off her 6pt, for a total of at least 15 move "portions," which leaves no room for a non-doublet in four rolls. Therefore, Blue started.
Blue played eleven crossovers and two spares off the 6pt, for a total of 13 "portions." One of them is a non-doublet (because we proved he started), and the other three rolls (requiring at least 11 portions) must all be doublets.
White's four doublets total 64 pips, Having two double 6s or two double 5s or 66 55 44 as part of the four-doublet combo would overrun the prime field. Other permutations can be eliminated either because of a similar overrun or because the opponent cannot block all the possible ways the back checkers might advance with certain doublets. The only combination that works for White is 66 55 33 22.
Blue has played 167 – 108 = 59 pips, so he has to have made an opening play of 3, 7 or 11 pips (so that 56, 52 or 48 pips is left over for the doublets). It is necessary to make the 7pt, else Opp can escape with her 66. As 33D will never be the best play early enough, and 11N leaves no way to make the prime with three other doublets, Blue's 7pt-making move must be opening 61P.
Here, through a careless error in logic, I managed to conclude that 61 55 44 44 is Blue's only workable combination of rolls, and that led me to believe that there are only two perfect-play solutions. To eliminate 61P-22N-55P-66O-44A-33O-44P-55J, I added the admittedly strained condition that all moves must be best according to all of XGR, XGR+ and XGR++ evaluations (though this also helped support the beneficial claim that all plays and cube actions were comfortably best -- no would-be-solver would need to consult a bot).
Given that 61P-22N fails the XGR test (heh heh), I thought I was down to one solution, which Ben and David both found: 61P-66O-55P-22A-44A-55P-44P-33O...
However, Robert conjured 61P-66O-55P-22A-66O-55P-22A-33O, and using the same Blue combination of 61 66 55 33, I discovered two additional solutions (transpositions), which are 61P-66O-55P-55P-66O-22A-22A-33O, and 61P-66O-66O-55P-55P-22A-22A-33O.
So, while a number of readers (including, no doubt, some that didn't respond) found this to be an enjoyable and/or challenging problem, it is not up to the standard that I would like. Fortunately, there are various ways to streamline the five solutions (including the one with 22N I discarded using XGR) down to one solution. For example:
......(a) The sequence cannot end with 33.
......(b) 66 cannot be the second roll of the sequence.
......(c) 55 cannot be the third roll of the sequence.
......(d) One player matches the roll of the other player thrice in a row.
For condition (a) or (b), the solution is uniquely 61P-22N-55P-66O-44A-33O-44P-55J. For condition (c), it is 61P-66O-66O-55P-55P-22A-22A-33O. For condition (d), it is 61P-66O-66O-55P-55P-22A-22A-33O.
If I reprint the problem at some point, I'll use one of the lettered conditions above or I'll figure something else out (e.g., altering the position).
................Nactation of the sequences
For 44 played 13/5(2), and for 55 played 13/3(2), I elected to use the style alternative of P (Point), but I could just as easily have chosen the areal alternative of O (Outer). Had I done so, then all doublets would have been areal, in fact "bi-areal" (played in two areas of the board). For an "areal overview," see section 5 (page 20–21) of the Nactation tutorial.
Moreover, the doublets moves in these sequences are simple doublets, meaning that they are played in straightforward pairs. Simple doublets (which sport the letters in the mnemonic "BEACON") are explained and illustrated in section 3 (page 9–11).
There is one exception, which is illustrated below.
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White's position before playing 33O | |||||
Just prior to the final position, in most of the sequences White reached the position above, and played 33O. "O" means that half of the move is played into (to or within) her Outer board, and the other half is played out of her Outer board. This is true even when doublets are not played in straightforward pairs.
To see this, break down the move into its four "portions" (sub-moves): 13/10 10/7 7/4 8/5. Those first two portions (13/10 and 10/7) are played into her Outer board, and the other two portions (7/4 and 8/5) are played out of her Outer board. That is all an O move is.
[A non-doublet O move is even more easier to understand: only one portion into the Outer board and one portion out of the Outer board. An example appears in section 4, page 13.]
I'll be happy to answer any questions.
Nack