Against a five-point board, White is about 30% to enter every time he rolls. Once he is moving, it will be hard for Black to clear the 9pt safely. As things stand, Black has 11 22 33 55 66 65 63 53, just 11 rolls, that do the job. But Black does not need the 9pt, because he is not trying to block White. He is just trying to get home safely. Should Black begin clearing the 9pt now?
In this problem, he has the opportunity. Three rolls (65 63 53) allow him to do so without leaving any fly shots. In addition, with 32, Black could lift the 9pt, and leave a blot on the 7pt, subject only to a 17-to-1 shot. With 52, Black could switch from the 9pt to the 7pt, giving a similar shot at the blot he leaves behind. Because of the other outside blots, even these risks might be too much. As soon as White hits any of Black’s blots, the threat of a gammon loss will loom large.
Alternatively, Black might try augmenting his blockade. With a made bar point, for instance, Black can add 62 52 32 to the list of rolls that clear his 9pt. When he owns the 8pt, Black has 61 51 31. But owning the 7pt or 8pt is problematic. Anytime Black makes another outside point, he may find it hard to clear that point later. When he goes to clear the 8pt, for example, Black will have 1 and 3 as “trouble” numbers. Clearing the 7pt is easier; the only trouble number is 2. Making the 10pt is wholely counterproductive. Owning it only increases the chance of blotting later.
In
Part (a), there is a subtle advantage to making a blocking point. Each blocking point gives a boost to the chances that White might enter, and be forced to break his perfect board. 51 and 54 are already cracking numbers. With a made bar point, White cracks on 52, as well. When Black owns the 8pt, White cracks with 53.
- 65 — 13/7, 12/7 ... Make a blocking point.
- 63 — 9/6, 9/3 ...... Clear the 9pt safely.
- 54 — 13/8, 12/8 ... Make a blocking point.
- 53 — 9/6, 9/4 ...... Clear the 9pt safely.
- 52 — 13/6 ........... Pay later.
- 32 — 13/11, 12/9 . Leave just 1 blot, and 2 shots.
In
Part (b), I’m going to clear the 9pt when I can do so safely. Otherwise, I’ll pay later, except with 32, where there is no safe play.
- 65 — 9/4, 9/3 ...... Clear the 9pt safely.
- 63 — 9/6, 9/3 ...... Clear the 9pt safely.
- 54 — 13/4 ........... Pay later.
- 53 — 9/6, 9/4 ...... Clear the 9pt safely.
- 52 — 13/6 ........... Pay later.
- 32 — 9/7, 9/6 ...... Cross my fingers, and bank on the return shots.
This is a fascinating problem. As is so often the case when I write a longer response, I am not at all confident about any of my answers. In Part (a) especially, I may have talked myself into folly by not simply clearing the 9pt whenever possible. I also wonder whether I should ignore the gammon risk, and switch points with 52 in one or both parts.
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 53%.
Two Plus Two
Re: Problem of the Week #136: January 29
Linear Mode
