Pay Me Now or Pay Me Later?
In their popular work
New Ideas in Backgammon (Woolsey and Heinrich, 1996), the authors suggest that at least two of the following three conditions should exist before is correct to pay now.
- If you pay now and get away with it, will the position be a near claim, or will there still be a lot of work to do?
- Is your opponent’s position improving, or is it likely to get worse in the next few rolls?
- Is your own position deteriorating quickly, or do you still have some flexibility?
In this position, the answers to the last two questions are relatively unambiguous. White is getting better, and Black is about to crack. The answer to the first is less certain. Not only might Black leave another shot next time, he might also lose the race. Nevertheless, it looks like “paying now” is the way to go.
What Happens When Black Pays Later?
The only safe play with this 43 that allows Black to keep his five-point board is 6/2, 4/1. After that, there is no more slack. On his next turn, Black will have to abandon his anchor or else break his six point. In the meantime, White has all the time in the world. He can play the four checkers on his eight point while he waits for Black to crash. Suppose Black were to play safe, and White, on his turn, did the same. What will happen on the 21 rolls awaiting Black after that? Surprisingly, there are only eight numbers that will force him to leave a blot. Here is the breakdown:
- Play inside: 16 (no six)
- Break 6 point: 11, 12, 13, 14, 15, 23, 24, 25, 34, 35, 45
- Run safely: 22, 33, 44, 55
- No move: 66
- Run and leave blot on 18 point: 26, 36, 46, 56
Unfortunately, White will continue to wait him out. In many variations, White can play again from his eight point when he rolls big numbers, and play inside with the small ones. When he rolls doubles, White can run from the rear. After that, playing safe will not be so easy for Black. His 6 point will be gone, so many more rolls will force him to play off his 18 point, and to leave shots.
These considerations make it clear. In all three parts of this problem, Black should pay now.
Leave Two Blots or One?
Having decided to run off his anchor, Black can play 18/15, 18/14, leaving two blots and 21 shots, or else he can run one checker all the way to safety, 18/11, leaving one blot and 24 shots. The former is generally the better play. It leaves fewer shots, and also makes it easier to clear the blots when they are not hit. Two checkers on the 14 and 15 points can be made safe with all but ten numbers (22, 33, 23, 24, 25 and 26). Of these, only 33 leaves more than the minimum 11 shots. When Black leaves a blot on the 18 point, however, he will have 11, 22, 33, 66, 12, 13, 14, 15, 16, 23 and 24 as trouble numbers. That’s 18 rolls in all, and most of them leave more than 11 shots. A few will force him to leave the same double shot that his opponent has just missed. If you crunch all these numbers, you will find the chance that Black will be hit after this turn or the next to be:
P( Black is hit in next two turns after playing 18/14, 18/15 ) = 62%
P( Black is hit in next two turns after playing 18/11 ) = 73.5%
Gammon Considerations
The gammon chances here are entirely one-sided. With his five point open, and little prospect to make it, Black does not figure to win any double games. But he does not figure to lose very many either. Unless Black gets both of his rear checkers closed out, the gammon chances in this position seem close to nil. With two Black checkers closed out, however, and an optimal distribution of spares, White is rated to win a gammon about 42% of the time.
Let’s guess that when Black runs with one checker something like 1% or 2% of all games end with him losing a gammon. When he runs with two checkers, thus leaving both vulnerable, we can add, say, 5% to 8% to these estimates.
Double Match Point
At double match point, the decision is easy. Black should play the percentages, without regard to gammon danger or how many blots may be exposed. Moving 18/14, 18/13 is the way to go.
My solution in Part a:
18/14, 18/13
Centered-Cube in Money Game
When gammons matter, and all other things are equal, it is true that leaving two blots is worse than leaving one. Can this tip the balance when White has access to the cube? Consider the sequence: Black runs, White hits, Black dances, White doubles. Might running one checker all the way be better than running both when White has access to a centered cube?
Not really. When Black is on the bar, and White is shooting a double shot to cover on his four point, Black must pass. It does not matter whether he has a blot floating in the outer board or not. It only makes a difference in the small number of cases where White can both hit, and lift his blot off of the four point. Without a second target for White to shoot at, the open four point allows Black to take. So, at the cost of 24 immediate shots instead of 21, and 18 shots next time instead of 10, Black could run all the way now, and assure himself of a take in the event of a big parlay. The parlay: White hits, lifts his blot, Black fails to enter, and White then doubles. The cost is too high.
When the cube is still centered in a money game, Black should run both checkers, 18/14, 18/13. The good news is that he won’t be gammoned. The bad news is that he must pass if he is doubled after a dance.
My solution in Part b:
18/14, 18/13
Black Owns Cube in Money Game
This is the tricky one. When Black owns the cube, gammons are active. In an attempt to lose fewer gammons, Black might choose to play 18/11 rather than 18/14, 18/15. His gammon price is 2 to 1. For every two gammons he saves, he can lose one extra game, and still break even.
I am not really certain whether he has that price. It’s very close. In part because of the quiz factor (QF), I will guess that he does. After all, it wouldn’t be very nice for Mr. Robertie to specify three conditions that were irrelevant to the answers!
My solution in Part c:
18/11
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 49%.