When Black plays safely in this position, it is pretty easy to narrow his options to just two. That is because Black cannot play a 4 from his 6pt. His only choices for the 4 are to play from his 4pt or 5pt. The only two safe plays are 5/off and [4/3 4/off].
Between the two, my first instinct is to lift the 4pt. It begins the bear off, and also leaves a flexible arrangement of checkers for the future. Black will be completely safe on the next turn, and probably for the turn after as well. By the time he has to leave a shot, if ever, he should have five or six checkers off. Even if he is hit, against White’s crashed board, he will be a big favorite.
The other safe alternative leaves a much more brittle position, both stripped and stacked. On the following turn, a roll of 64 will force a blot, and eight others (65 62 53 42) fail to bear off any checkers. 44 opens up a double gap.
If those were his only choices, it is an easy call. Clearing the 4pt is much the better of the two.
What about leaving a shot? Can Black increase the rate at which he wins gammons when he rips off two checkers now? If he does pull two, he will have 13 crossovers remaining. White will need 8 to beat the gammon. By itself, this does not seem to give any motive for taking a risk. On the other hand, there is little reason not to. The 2 that White would use to hit is duplicated. White also needs a 2 (or 1) to safety the exposed blot in his home board. Furthermore, unless White hits with a roll of 22, he will leave many return shots.
Let’s count the return shots when White hits with each of his 2s. Although hitting may not be best, here are the best plays for White when does hit:
- 21E (Each) = 23/21*, 3/2 — Black has 32 return shots (any entering roll)
- 22R (Run) .= 23/21*(2), 12/10, 3/1 — Black has 1 return shot (55)
- 23U (Up) .x= 23/21*/18 — Black has 23 return shots (any 1, 3 or 4 except 11 12 24, plus 66)
- 24R (Run) .= 23/21*/17 — Black has 22 return shots (any 1, 3 or 4 except 11 12 24)
- 25R (Run) .= 23/21*/16 — Black has 22 return shots (any 1, 3 or 4 except 11 12 24)
- 26R (Run) .= 23/21*/15 — Black has 23 return shots (any 1, 3 or 4 except 11 12 24, plus 55)
Once the hitting begins, Black has a chance to suck up all of White’s blots, and coast home to a gammon victory.
My solution: 4/off, 1/off
For the Record
I am so often wrong that I like to post my record in these messages. It's kind of a truth-in-advertising thing. Grunch
: I have been answering these problems without the use of a bot, and before checking the excellent solutions of others, since Problem 28. My record at this writing is 54%.