Here’s a problem of a familiar sort. Black has been steadily building his front position, while White has tossed in a 5-3 and a 6-5. White’s last roll of 6-5 completes the escape of his back checkers, but leaves him with a huge stack on his midpoint. Now that White has escaped entirely, Black has to leave his front position alone and start mobilizing the back men. He has two choices: 24/20 11/10, or 24/23 13/9. (24/23 11/7 is all right as well, but 13/9 yields a slightly more active distribution.) Which is best?
Years ago, most strong players would have played 24/20 in a shot. White must start moving the checkers on the midpoint, and this play guards the outer board. It also prepares to anchor on the 20-point, which is key since the race isn’t hopeless yet (Black trails by 8 after the roll.) And the play has no obvious downside. True, there’s nothing really wrong with 24/23, but the play doesn’t guard the outer board and doesn’t prepare to make an anchor, so there’s no real reason to choose it. Or is there?
The bots play 24/23 13/9. Rollouts support this choice and in fact indicate that plays involving 24/20 are not just mistakes, but blunders! How can we explain this?
Most middle game backgammon problems can be solved by resorting to a small set of relatively simple ideas: if you can’t do something big like making points or hitting blots or building a prime, try to arrange your checkers efficiently so that you’re preparing to do something good next turn at minimum risk. Here playing 24/20 prepares to make the 20-point while guarding the outfield. At the same time, White has only a handful of numbers that point on Black’s head. So 24/20 11/10 would seem to fit the bill perfectly.
But there are a few positions that just aren’t that simple. Sometimes we actually have to consider how our opponent’s rolls play. This sort of concrete analysis can yield some very surprising results. In this case, the difference between 24/20 11/10 and 24/23 13/9 comes from three groups of swing rolls.
Playing 24/20 11/10 gains strongly after two of White’s replies: 5-4 and 4-2. White will play 13/8 13/9 with a 5-4 after either play, but splitting to the 20-point gets Black a direct shot instead of an indirect. 4-2 is a little different. After 24/23. White will play 13/9 13/11 with a 4-2, yielding some indirects. After 24/20, White will revert to making the 4-point, but Black will get a direct shot again. In each case, the blot on the 20-point forces a direct shot, as the play was designed to do.
Playing 24/23 gains modestly on the doubles. Black doesn’t get pointed on after 1-1, 3-3, and 4-4, which is a solid plus. After the big doubles (6-6 and 5-5), the race goes in the crapper, so it’s an advantage to be able to anchor back on the 23-point rather than up on the 20-point.
The key non-doubles that favor 24/23 are 6-3, 6-2, 5-1, and 3-1. After 6-3, played 13/4 in either case, Black gets a double shot rather than a single. After 6-2 and 5-1, Black gets a double shot without being hit. And 3-1 obviously favors not being on the 20-point.
Other than these swing numbers, the remaining rolls are pretty much a wash. Splitting to the 20-point may force White to alter his play, but without any real gain.
This is in essence a problem in efficiency. Splitting to the 23-point does as much work in getting a shot as the more obvious 24/20, for less risk. But finding such a play over the board, for the right reasons, would be extremely difficult. After all, no one is standing by your side ringing a bell to alert you that this position doesn’t conform to general principles.