First of all, congratulations to Mike, Stick and Scotty, for their correct solutions to both puzzles!
PUZZLE 1
Find a sequence of perfectly played moves that reaches the position below in FOUR rolls. (Blue-White-Blue-White.)
There are three solutions to Puzzle 1 above:
.....#1..52S-63R-42@-63P
.....#2..51S-63R-43@-63P
.....#3..51S-43U-43@-65P
Alternate Nactations: First roll: 52S can be 52Z, or 51S can be 51Z (though not commonly). Third roll: 42@ can be 42U, or 43@ can be 43U. Fourth roll: 63R can be 63P or 63@, or 65P can be 65@ (or even 65R). For explanations, refer to the Nactation tutorial (and if you're still not sure, ask me).
Some of you noticed that on the final roll, 54 can be used instead of 63. However, the move is identical (24/15) and therefore it does not represent extra solution(s); 54 is only an alternate expression. Refer to the "Distinct Move Rule," which was stated parenthetically here, then expounded on here.
When I created Puzzle 1, I believed there to be only two solutions. For 52S-63R-42, from my own rule of thumb ("Either 5pt beats either 4pt") I knew the best play below according to rollouts is @ (anchor, 22/20 24/20) by about .012, and was surprised to see that the XGR++ evaluation showed P on top. I even did the usual triple check. Or at least that is how I remember it.
Due to a couple of other differences I'd encountered, I happened to do an XG update the next day (so I can no longer duplicate the overturn I was convinced was there), and now, as you can see, XGR++ gets the move right, even though the margin is well understated (a mere half millipoint of .0005, if a fourth decimal place is included).
| 1. | XG Roller++ | 24/20 22/20 | eq: +0.152 |
| Player:
Opponent: | 52.64% (G:11.12% B:0.25%)
47.36% (G:5.87% B:0.19%) | |
|
| 2. | XG Roller++ | 8/4 6/4 | eq: +0.151 (0.000) |
| Player:
Opponent: | 52.52% (G:15.24% B:0.44%)
47.48% (G:9.95% B:0.37%) | |
|
Therefore, 52S-63R-42@-63P is not only a best-play-worthy sequence to Puzzle 1 by rollout, but it is also a legitimate solution by XGR++ evaluation (the arbiter for these puzzles). But my thinking being otherwise at the time I initiated the thread, that threw off Mike and Stick, who naturally stopped looking after they found two solutions (though for Mike it was #1 and #3, for Stick it was #1 and #2). When I stated in a follow-up post that there are three, Mike e-mailed me the third solution. I'm certain Stick would have found it as well, given that solution #3 shares the same opening move and reply of Puzzle 2's sole solution (which he found).
Puzzle 1's solution #2 is similar, but along the way it involves a play that XG 3-ply gets wrong, and many (if not most) people would get wrong at the board. See diagram below.
| 1. | XG Roller++ | 24/20 23/20 | eq: +0.152 |
| Player:
Opponent: | 52.64% (G:11.12% B:0.25%)
47.36% (G:5.87% B:0.19%) | |
|
| 2. | XG Roller++ | 24/20 13/10* | eq: +0.142 (-0.009) |
| Player:
Opponent: | 53.09% (G:14.63% B:0.64%)
46.91% (G:10.93% B:0.44%) | |
|
| 3. | XG Roller++ | 13/9 13/10* | eq: +0.114 (-0.037) |
| Player:
Opponent: | 52.05% (G:16.25% B:0.77%)
47.95% (G:12.39% B:0.59%) | |
|
It is these sorts of plays that you will learn naturally as you solve retro-puzzles. In this case, the margin is only .009, but as you (inevitably) stumble on many larger ones, you will be able to correct moves that you would (or might) otherwise misplay the next time they arise in live play.
For example, let's look at the position on the front cover (and analyzed on page 44) of Backgammon Openings. If you would hit with 31 in the position below, you have a chance to correct a big blunder (margin of .088 when rolled out) the next time the position (or one like it) arises. Even if you believed hitting is wrong but you are surprised at how wrong it is, you will hone your intuition and be less likely to go astray in similar positions.
| 1. | XG Roller++ | 8/5 6/5 | eq: +0.226 |
| Player:
Opponent: | 54.42% (G:15.28% B:0.44%)
45.58% (G:8.91% B:0.32%) | |
|
| 2. | XG Roller++ | 24/20 | eq: +0.148 (-0.078) |
| Player:
Opponent: | 52.56% (G:11.09% B:0.24%)
47.44% (G:5.73% B:0.19%) | |
|
| 3. | XG Roller++ | 13/10* 10/9 | eq: +0.143 (-0.083) |
| Player:
Opponent: | 52.87% (G:15.36% B:0.64%)
47.13% (G:10.93% B:0.45%) | |
|
| 4. | XG Roller++ | 24/23 13/10* | eq: +0.142 (-0.084) |
| Player:
Opponent: | 53.09% (G:14.63% B:0.64%)
46.91% (G:10.93% B:0.44%) | |
|
Note, too, that the above position shows why 54S-63R-31@-63P is not (yet another) solution to puzzle 1. That is, Blue should neither hit nor anchor. [Also, 54S-63R-11@-63P fails, because instead of anchoring, double 1s should of course be played as N (Near, 8/7(2) 6/5(2)).]
By the way, in case you do not know (and hadn't checked XGR++), White misplayed her 54S-63 reply even to reach the above position. That situation is discussed in the preface of Backgammon Openings (linked above the diagram). Nor would 54S-54R qualify as a partial route, as White should make the mirror image S play (copying Blue) with her 54.
To wrap up this puzzle, let's look at what happens in solution #3.
If you didn't already know that the best move with 51S-43 is U (Up, 24/21 24/20), you might well have stumbled across it as you worked on solving this puzzle. White's move of U (Up, 21/20, 24/20) is indeed best, according to XGR++ evaluation (and rollout; e.g., [U D3 S7 Z12] "^46*32 14). Below, White's last (and only) move was 43U.
| 1. | XG Roller++ | 24/20 23/20 | eq: +0.138 |
| Player:
Opponent: | 52.99% (G:8.73% B:0.22%)
47.01% (G:5.35% B:0.18%) | |
|
| 2. | XG Roller++ | 8/5* 8/4* | eq: +0.051 (-0.087) |
| Player:
Opponent: | 50.72% (G:15.62% B:0.58%)
49.28% (G:13.70% B:0.72%) | |
|
After White's 43U, Blue has a 43 of his own to play. Should he hit twice or anchor? XGR++ provides the answer (above). One more roll for White after that, and solution #3 is complete, as White makes her 15pt (i.e., 51S-43U-43@-65P).
While you were here, did you happen to notice that the rolls of 32 and 21 offer Blue a similar choice of anchoring or hitting twice? Well, what should you do with 32? With 21? If you like, you can learn that supplemental but useful information by checking XGR++'s evaluations before getting back to putting together a solution. You may as well seek additional ways to improve your backgammon while you're having fun working on the puzzle.
PUZZLE 2
Find a sequence of perfectly played moves that reaches the position below in FOUR rolls. (Blue-White-Blue-White.)
This puzzle has a unique solution. The first two moves and the fourth move are the same as solution #3 of the first puzzle. For reference, the midway diagram is repeated below.
| 1. | XG Roller++ | 23/13 8/3(2) | eq: +0.267 |
| Player:
Opponent: | 56.67% (G:12.96% B:0.38%)
43.33% (G:8.05% B:0.28%) | |
|
| 2. | XG Roller++ | 23/3 | eq: +0.230 (-0.037) |
| Player:
Opponent: | 55.95% (G:11.75% B:0.35%)
44.05% (G:8.16% B:0.24%) | |
|
| 3. | XG Roller++ | 23/18 13/3 8/3 | eq: +0.216 (-0.051) |
| Player:
Opponent: | 55.31% (G:14.31% B:0.60%)
44.69% (G:9.60% B:0.35%) | |
|
| 4. | XG Roller++ | 8/3(2) 6/1(2) | eq: +0.195 (-0.072) |
| Player:
Opponent: | 53.39% (G:18.28% B:0.31%)
46.61% (G:10.20% B:0.45%) | |
|
| 5. | XG Roller++ | 23/8 13/8 | eq: +0.191 (-0.077) |
| Player:
Opponent: | 54.90% (G:11.28% B:0.31%)
45.10% (G:8.29% B:0.26%) | |
|
| 6. | XG Roller++ | 23/13 6/1(2) | eq: +0.190 (-0.078) |
| Player:
Opponent: | 54.64% (G:12.71% B:0.22%)
45.36% (G:8.79% B:0.28%) | |
|
This time Blue's roll is 55. The best play, as indicated above, is C (Cross, 23/13 8/3(2)). After Blue makes that move, White finishes by making her 15pt with 65P.
I had considered either replacing Puzzle 1 or adding a Puzzle 3, with Blue having a 53 above (instead of 43 or 55). First, let me say that I've seen this roll misplayed as H (Hit, 13/5*), which is a blunder. [Possibly it was confused with the second-roll scenario occasionally seen of 43U-53, where H rolls out slightly better than P, though counter to ++ evaluation!]
The correct move with 53 in the above position is P (Point, 8/3 5/3). When White ends the sequence with 65, it reaches the extra puzzle position below that I had in mind:
PUZZLE 3
(The puzzle that I did NOT include in the original post)
Find a sequence of perfectly played moves that reaches the position below in FOUR rolls. (Blue-White-Blue-White.)
As I mentioned, the solution is 51S-43U-53P-65P. Note that 51S-63R-53P-63P fails because Blue's 53P is an error of around .04. Any of three hitting plays with 53 is better (and you will likely find it educational to compare that situation in parallel with the final diagram of this post).
However, there is a legitimate second solution. ..Blue opens with 41S, and now:
| 1. | XG Roller++ | 24/15 | eq: -0.145 |
| Player:
Opponent: | 47.13% (G:10.66% B:0.46%)
52.87% (G:15.79% B:0.58%) | |
|
| 2. | XG Roller++ | 24/18 13/10 | eq: -0.154 (-0.009) |
| Player:
Opponent: | 46.89% (G:11.47% B:0.49%)
53.11% (G:16.19% B:0.68%) | |
|
| 3. | XG Roller++ | 24/21 8/2* | eq: -0.173 (-0.028) |
| Player:
Opponent: | 45.88% (G:11.97% B:0.47%)
54.12% (G:15.75% B:0.76%) | |
|
In XG 3-ply, S (Split, 24/18 13/10) edges out R (Run, 24/15) by .001. And by rollout (not shown), S beats R by .002 or .003. However, as you can see above, XGR++ (evaluation) prefers R by .009. Thus R (though beaten in the rollout) is the move that qualifies for a solution. While I don't deem rollout overturns to be an automatic reason for exclusion, it is enough of an imperfection that I elected to exclude this puzzle from the original post.
I bring this up for a few reasons:
(a) So that you know that 41S-63S is actually best, according to rollout.
(b) To remind you that XGR++ is typically the arbiter of my puzzles; this relieves you (the solver) of the burden of having to roll out positions or find reference rollouts.
(c) So that the process of solving a puzzle does not cause you to learn an opening play or reply from an XGR++ evaluation that disagrees with its best available rollout.
(d) To make you aware that if I decide to include such a puzzle anyway, I will inform you (in illustrated fashion) of any known ++/rollout discrepancy in the solution post.
(e) As a further opportunity to mention that instead of "perfect," I might occasionally choose to announce the criteria as "very well played" (within .01) or "well played" (within .02). In recapping solutions, I will always let you know when a non-optimal play is part of it (and what the optimal play would have been).
Continuing from the previous diagram...
| 1. | XG Roller++ | 9/3 6/3 | eq: +0.071 |
| Player:
Opponent: | 50.63% (G:14.37% B:0.40%)
49.37% (G:10.15% B:0.42%) | |
|
| 2. | XG Roller++ | 24/18 13/10* | eq: +0.065 (-0.007) |
| Player:
Opponent: | 51.58% (G:14.07% B:0.63%)
48.42% (G:12.89% B:0.59%) | |
|
| 3. | XG Roller++ | 13/7 13/10* | eq: +0.018 (-0.053) |
| Player:
Opponent: | 49.95% (G:15.63% B:0.75%)
50.05% (G:14.21% B:0.89%) | |
|
In his typically humble fashion, Mike voluntarily confessed (after getting both solutions to puzzle 3, which I had e-mailed him as a bonus), that before working on this puzzle, he would have played X (hit-and-split, 24/18 13/10*) over the board. So would I have, the first time I ran across it. (So would XG 3-ply, by the way, and by .024.) But he plugged it into XGR++ anyway to make sure, and was surprised. Mike learned something here, and I very much doubt he is alone.
Going forward, after Blue makes his best move of P (Point, 9/3 6/3), White finishes with 63P (Point, 24/15), and the puzzle #3 position is reached.
Nack
