If volatility is what forces or restrains a double, then this position is a dream. Whoever wins is going to pocket a passel of gammons. Owning the cube in this sort of double-edged position is a huge asset, and often it is correct to keep it if you have it. Doubling your opponent out – rather than in – is the winning line in a prime-versus-prime game.
So, if the rule is to hold onto the cube in a timing battle, is Problem 86 an exception? My first instinct was emphatically, “No!” After a second look, however, I began to see how strong Black’s position is. Any roll with a 1 will allow him to clean up both home board blots, and there are a few others that will do the same job even better. In fact, there are only 17 rolls that force Black to leave an exposed checker in his inner board. And there are no rolls that force him to leave two, although he might choose to voluntarily. When it comes to the blitz, Black can add at least one point to his inner board on this turn. With 5 rolls, he can add two.
Without addressing the issue of voluntary expose, let’s assume for the moment that Black decides to “play safely,” covering as many inner board blots and points as he can. Then the numbers work out like this:
- Black makes a four-point board with 31 rolls, and a five-point board with the other 5.
- Black has 17 rolls that leave a blot in his home board. The other 19 do not.
- White can enter both checkers against a four-point board with 4 rolls. He can enter exactly one with 16 rolls. He dances on the other 16 rolls.
- Similarly, against a five-point board, White enters both checkers with just 1 roll, he enters exactly one with 10 rolls, and he fans with the other 25.
- Combining these chances, we have these odds in 1296 rolls: White enters both checkers 10% of the time (129/1296), he enters exactly one 42% of the time (546/1296), and he fans a whopping 48% of the time (621/1296).
- White’s chance of hitting in 1296 rolls is only 14% (187/1296).
Is this sounding like a pass to you? It sure does to me. The real issue is whether Black is too good to double. The only thing that might induce Black to cash now is the two-way gammon threat. Everything could change in one roll. Perhaps this situation fits the the prime-versus-prime rule given above after all. If Black wants to, he can double his opponent out, just as the rule advises.
The rolls with a six present Black with an interesting choice. Should he cover one of his blots (usually on the one point), or should he escape his rear checker? With 6-1 and 6-6, of course, he gets to do both. On a 6-5, he should cover twice inside, making a five-point board. The alternative, 24/18, 5/2 is not bad either, but why leave a shot against White’s five-point board? That leaves the rolls of 6-4, 6-3 and 6-2.
- Black rolls 6-4: The “safe” play, referred to above, is 7/1, 18/14. Black risks 11 shots by White, keeps his five-point prime, adds an inner board point, and moves into position to hit a third White checker. If Black chooses to escape instead, he will probably open up outside, playing 24/14. This leaves four blots around the board, and gives White a double direct shot. Not much of an alternative. The other sixes are harder.
- Black rolls 6-3: Once again, the safe play covers on the one point, 7/1, 18/15. Black again risks 11 shots, keeps his five-point prime, adds an inner board point, and moves into position to hit a third White checker. All the same good things as before! Against this, Black has the alternative of escape. He can play 24/18, 7/4, which leaves only the two inner board blots. Even in the worst case, when White hits Black and closes him out, White very well may crack before he escapes his own rear checkers. It’s a close call.
- Black rolls 6-2: Here, Black has three choices. Thematically, making the one point, 7/1, 18/16, has the same advantages as with 6-4 and 6-3. The outside blot, however, duplicates Black’s own escape number. He needs a six to escape, and he’s giving himself a six to hit the third checker. The second alternative, 24/18, 7/5, also repeats the theme from above. Black escapes, while leaving only the two blots. The third alternative, 13/10*, is an entirely different animal. When Black hits the third checker, he gives himself two winning game plans. First, White might miss the shot from the bar, and be closed out in the ensuing blitz. Second, when White hits, Black may not care. White seems certain to crack before he can escape three checkers. In fact, hitting hurts White in a way, because in many cases White’s checker will be primed on the one or two point after a hit. White might prefer to enter on the three point.
With 6-4 and 6-3, my inclination is to cover inside, but with the 6-2, the immediate chance of White cracking makes me tend toward the timing play. One final note on these rolls with a six: Black should avoid breaking the eight point to cover on the the two. He needs his prime in case he is hit.
Of the other rolls Black might get, only 4-3 and 5-4 present any real choices. Both can be used to cover inside, by abandoning Black’s prime. With 4-3, Black can play two checkers off the five point. On a roll of 5-4, he can do the same thing from the six.
- Black rolls 4-3: Here, the choice is not hard. The alternative is to play 8/1, giving up the five-point prime, while still leaving an inside blot. Black should play safe instead.
- Black rolls 5-4: The alternative with this roll, 7/2, 18/14, is not as weak. Black can cover on the two point, while keeping his outside blockade. A third possibility, making the three point, should be quickly discarded. It leaves a double direct shot, while giving White the chance to both hit and position himself for escape on the same shot.
In these cases, I am attracted to the safe play. White’s five point board inspires a certain respect (if no real fear), and his third checker waiting to be hit outside gives me just the bait I need to justify a pick-and-pass strategy. Besides, look how good the numbers are when Black plays safe!
My Solution
Part (a)
Double/pass
Part (b)
(1) 6-2:
18/10*
(2) 6-3:
7/1, 18/15
(3) 4-3:
5/1, 5/2
(4) 5-4:
6/1, 6/2
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 50%.
Last edited by Taper_Mike; 12-06-2010 at 02:06 AM.
