Two Problems Posed by Nack
For anyone that would like a tough challenge (or two):
(1) How can the above position be reached with perfect play from the opening position? (Any number of rolls is allowed. I believe the fewest is six.)
(2) How can you get from this position to Snua's position at the top of this post? (The sequence I found is nine rolls.)
I am not sure I can solve part (2), but I had a fun time solving part (1). As so often happens, I discovered several interesting positions along the way. I’ll show a contrasting pair at the end of this post. Solving retro problems is a great way practice your early game. You run through a ton of positions where you are challenged to find the best play. The two I am posting below are a pair I encountered while solving Nack’s challenge.
Nack gave a big hint when revealed that a six-move solution exists. Since White is on roll after the sixth move is played, that means White must have rolled first. White had the 1st, 3rd, and 5th rolls; he is now on roll on the 7th.
Since each player only gets three moves, I made the assumption that most moves took checkers to their final destinations. On the opening roll, in particular, I assumed that White played either 32S = 24/21 13/11 or 62S = 24/18 13/11. Having done a few of these retro puzzle before, I knew that those are the only perfectly played opening moves that put a checker on the 11pt.
Now, 32S = 24/21 13/11 looks especially promising. It puts two checkers exactly where we want them. That was the play I tried first.
Since Blue must, at some point, play a checker down from his midpoint, and have that checker get hit, I figured this was as a good a time as any. I tried 63H = 54H = 13/4*. That leads to the following position.
         
 
         
 | | White is Player 2
score: 0
pip: 166 | Unlimited Game
Jacoby Beaver | pip: 158
score: 0
Blue is Player 1 |
|
| XGID=aa--A-E-C---dDa--c-e----B-:0:0:-1:43:0:0:3:0:10 |
| White to play 43 |
Being not quite as thick as a brick, I saw right away that 43@ = bar/21* 24/21 would put White’s rear position in order. In fact, on his next turn, all he would need is a 31 to reach the desired final position.
Next came the fun part. Blue is on roll, and I had some trouble with the position below. What did he roll?
 
| | White is Player 2
score: 0
pip: 159 | Unlimited Game
Jacoby Beaver | pip: 179
score: 0
Blue is Player 1 |
|
| XGID=----b-E-C---dDa--c-e----BA:0:0:1:00:0:0:3:0:10 |
| Blue on roll, cube action? |
I toyed with idea of having him dance, but that did not work. Dancing would leave him with only one more move, and three checkers that had to move. One move would be used to enter, a second to split the 24pt, and a third to play down to the 8pt. This convinced me that Blue did not dance.
The next thing that became obvious was that Blue had to get a checker to the 20pt. I knew that because I already had White’s next move. It was 31P = 8/5* 6/5. Blue is on the bar in the final position. If he enters on this turn, he must somehow get hit. The only way that is possible is if he moves up to the 20pt.
Pretty simple. But it still took me a couple of false starts before I stumbled onto 42. At first, I tried 51U = bar/20 24/23, but a quick check with XGR++ showed that U is not the best play. Eventually, I saw that 42U transposes with 51U. 42U = bar/23 24/20, a play that puts checkers in the same places as 51U.
| | White is Player 2
score: 0
pip: 159 | Unlimited Game
Jacoby Beaver | pip: 179
score: 0
Blue is Player 1 |
|
| XGID=----b-E-C---dDa--c-e----BA:0:0:1:42:0:0:3:0:10 |
| Blue to play 42 |
| 1. | XG Roller++ | Bar/23 24/20 | eq: -0.3768 |
| Player:
Opponent: | 41.28% (G:8.80% B:0.27%)
58.72% (G:16.32% B:0.75%) | |
|
| 2. | XG Roller++ | Bar/21 24/22 | eq: -0.3796 (-0.0028) |
| Player:
Opponent: | 41.29% (G:8.73% B:0.29%)
58.71% (G:16.77% B:0.78%) | |
|
| 3. | XG Roller++ | Bar/21 13/11 | eq: -0.4205 (-0.0436) |
| Player:
Opponent: | 40.73% (G:8.90% B:0.31%)
59.27% (G:17.91% B:1.14%) | |
|
eXtreme Gammon Version: 2.10
The rest was easy. White points with 31P = 8/5* 6/5. Blue dances. White cubes. The cube occurs on the 7th move. The position is reached in 6 moves. The complete sequence is: 32S-63H-43@-42U-31P-F-C.
At this point, I was quite pleased, and began writing up the solution. I wrote all of the above, in fact, and I was just about to click “Post Message,” when the roof caved in. My solution was wrong! A last-minute check showed that it puts only three checkers on Blue’s 8pt. Oh, man!
Desired Position
     
| | White is Player 2
score: 0
pip: 155 | Unlimited Game
Jacoby | pip: 173
score: 0
Blue is Player 1 |
|
| XGID=----b-E-D---dCa--b-db--AAA:0:0:-1:00:0:0:1:0:10 |
| White on roll, cube action? |
32S-63H-43@-42U-31P-F-C
| | White is Player 2
score: 0
pip: 155 | Unlimited Game
Jacoby | pip: 178
score: 0
Blue is Player 1 |
|
| XGID=----b-E-C---dDa--b-db--AAA:0:0:-1:00:0:0:1:0:10 |
| White on roll, cube action? |
I took a short break, and then took another look. One easy way for Blue to get a fourth checker on his 8pt was to have him play 51S, 52S, or 54S somewhere along the way. Perhaps one of those was his opening roll. Note that all three are splitting plays where the small number (1, 2, or 4) is used to split off the anchor, and the 5 is used to play down from the midpoint, 13/8.
Happy with the moves I had created for White, my first attempt was to add a 51S at the beginning. Since I was letting Blue move first, that meant that my solution would be seven moves. I was okay with that. At least it would be a solution. It did not take long to discover that was unworkable. Blue had too much freedom. I could not find a way to make him hit on the 4pt. In fact, if Blue opens 51S, and White follows with 32S, then Blue should hit outside when he rolls a 63 on the third roll.
| | White is Player 2
score: 0
pip: 162 | Unlimited Game
Jacoby Beaver | pip: 161
score: 0
Blue is Player 1 |
|
| XGID=-a--a-E-D---dDa--c-e---AA-:0:0:1:63:0:0:3:0:10 |
| Blue to play 63 |
That taught me something. In order to prevent Blue from hitting outside, White needs to make his 5pt beforehand. What if I had White open with 31P? Blue could play his 51 on the second roll. It might work. Plus, it would get me back to a six-move solution.
In short order, I cooked up a solution: 31P-51S-32S-63H-43@-F-C. I was just about to write that one up, when I discovered a problem with the 32S on the third roll. 31P-51S-32 is an early position that reminds me of 51S-41. The best play is the beginner’s safe play! Once again, I was learning about early-game play as I tried to solve a retro puzzle.
 
| | White is Player 2
score: 0
pip: 163 | Unlimited Game
Jacoby Beaver | pip: 161
score: 0
Blue is Player 1 |
|
| XGID=-b----E-D---eD---b-db--AA-:0:0:-1:32:0:0:3:0:10 |
| White to play 32 |
| 1. | XG Roller++ | 13/8 | eq: +0.2428 |
| Player:
Opponent: | 55.30% (G:16.92% B:0.62%)
44.70% (G:11.39% B:0.53%) | |
|
| 2. | XG Roller++ | 24/21 13/11 | eq: +0.2296 (-0.0132) |
| Player:
Opponent: | 55.34% (G:15.61% B:0.63%)
44.66% (G:11.15% B:0.44%) | |
|
| 3. | XG Roller++ | 24/22 13/10 | eq: +0.2145 (-0.0283) |
| Player:
Opponent: | 55.02% (G:16.04% B:0.67%)
44.98% (G:12.08% B:0.47%) | |
|
| 4. | XG Roller++ | 13/11 13/10 | eq: +0.2032 (-0.0396) |
| Player:
Opponent: | 54.61% (G:17.12% B:0.72%)
45.39% (G:12.44% B:0.69%) | |
|
eXtreme Gammon Version: 2.10
Okay. If that was not going work, I would go back to opening with 32S. I already knew that White had to make his 5pt before I gave Blue a 63, so I tried 32S-51S-31P as the first three moves. Now, I was ready for Blue to roll 63.
| | White is Player 2
score: 0
pip: 158 | Unlimited Game
Jacoby Beaver | pip: 161
score: 0
Blue is Player 1 |
|
| XGID=-a--a-E-D---dDa--b-db--AA-:0:0:1:63:0:0:3:0:10 |
| Blue to play 63 |
| 1. | XG Roller++ | 13/4* | eq: -0.4514 |
| Player:
Opponent: | 39.53% (G:10.26% B:0.45%)
60.47% (G:19.43% B:1.03%) | |
|
| 2. | XG Roller++ | 24/15 | eq: -0.5164 (-0.0650) |
| Player:
Opponent: | 38.36% (G:8.22% B:0.32%)
61.64% (G:20.52% B:0.75%) | |
|
eXtreme Gammon Version: 2.10
Once again, the rest was easy. White anchors with 43@ = bar/21* 24/21, Blue fans, and White cubes. The final sequence was:
Challenge 1 Solution
32S-51S-31P-63H-43@-F-C
Two Early-game Positions
Now take a look at the 4th-roll positions I discovered along the way. Both have been added to my database. Actually, one of them was already there, but I had forgotten it.
How do you play these?
Mike
