Cube Handling in Races

Can you take a double in a symmetric position?

Position A.

Position B.

Position C.

As Positions A, B, and C show, the answer is (A) yes, (B) no, and (C) maybe. Surprisingly often, though, the answer turns out to be yes.

A one-roll symmetric position is only a take if you have between eighteen and twenty-seven rolls that bear off. These are the only one-roll symmetric takes:*

They are only takes because of your redoubling equity. In a one-roll symmetric position your winning chance, if the game is played to the end, is less than twenty-five percent, but the cube makes up the deficiency in these positions.

Position D.

In a two-roll position, like D, the decision to take depends on your chances of bearing off in one, two, and three rolls. In Position D, your chance of bearing off in one roll is 2/36 (5-5 or 6-6). Your chance of needing three (or more) rolls to bear off is 685/1296, or about 19/36. Your chance of bearing off in two rolls is 15/36. Overall, your chance to win, ignoring the cube is:

	34 36  ×  2 36	(large doubletons win)
+	19 36  ×  15 36	(your opponent misses, you don't)
=	353 1296	= .27, a take.

(Your actual chances are better, because your rolls of 2-2, 3-4, 3-5, 3-6, 4-5, 4-6, and 5-6 will give you a powerful redouble.)

In such two-roll symmetric positions, you can usually tell whether to take a double if you know three numbers. Let A be the chance you are off in one roll (usually large doubletons), B the chance you are off in two rolls (found by examining all possible two-roll parlays), and C the chance that you will need three or more rolls. In Position D, A is 2/36, B is 15/36, and C is 19/36.

You can always take if A² + B² + C² < 1/2. In Position D, A² + B² + C² = (4 + 225 + 289)/1296 = 518/1296 = .40, well short of 1/2.

Here are some positions that are takes, by the A² + B² + C² < 1/2 criterion:

Position E. A = 2/36, B = 17/36, C = 17/36. A2 + B2 + C2 = 582/1296, a take.

Position F. A = 2/36, B = 19.5/36, C = 14.5/36. A2 + B2 + C2 = 594.5/1296, a take.

Position G. A = 5/36, B = 16.6/36, C = 14.4/36. A2 + B2 + C2 = 508/1296, a take.

These, however, are clear drops.

A = 2/36, B = 29/36, C = 5/36. A2 + B2 + C2 = 1070/1296, much larger than 1/2, a drop.

A = 4/36, B = 29/36, C = 3/36. A2 + B2 + C2 = 866/1296, a drop.

A = 3/36, B = 27/36, C = 6/36. A2 + B2 + C2 = 819/1296, a drop.

In case you don't want to evaluate A² + B² + C² in your head over the board, a rule of thumb deals accurately with almost all cases:

A = 2/36, B = 23/36, C = 11/36. A2 + B2 + C2 = .504; Black's redoubles give him a take.

A = 3/36, B = 24/36, C = 9/36. A2 + B2 + C2 = .514; Black's redoubles give him a bare take. (To drop gives up an equity of two percent of the cube.)

Looking back at Position D, you could say, without doing the calculation exactly, that all three numbers must be well under 2/3, and by the rule of thumb the position is a take.

The position above is a well-known proposition. Each player's chances to be off in two rolls, three, and four or more, are respectively 3/36, 12/36, and 21/26. A² + B² + C² = 594/1296 = .46, a clear take.

Positions where neither player can miss are only takes if there is redoubling equity.

Position H.  A drop.

A take.

In such positions, where a doubleton gives you a good redouble, you must still have a no-cube winning chance of at least 3/16 (.1875) to take the cube. Position C is the only possible position for taking with a chance of exactly 3/16; as a rule of thumb, always drop if your no-cube chance is less than 23% (i.e., your chance for a helpful doubleton is less than 36%). In Position H, your chance to go off in three rolls or less is .42; your winning chance is .42 × (1 − .42) = .243, which, when supplemented by a little redoubling vigorish, gives you a take.

In Position I, however, neither 1-1 nor 2-2 help you; your chance to go off in three rolls or less is about .30, your winning chance is .3 × (1 − .3) = .21, and you should drop.

Position I.

Position J.

In Position J, your chance to go off in three rolls or less is about .36, and your no-cube winning chance is .36 × .64 = .230. By redoubling you also win in those games where your first roll is a doubleton, and White's third roll would have been. This gives you an additional chance of about

	31/36	(White starts with a singleton)
×	5/36	(Black rolls a useful doubleton)
×	31/36	(White rolls another singleton)
×	31/36	(Black rolls a singleton)
×	5/36	(White bears off with a doubleton)
=	.012.

Finally, you may roll two doubletons to your opponent's one, with a chance of (5/36)² × 2 × (5/36) × (31/36) = .005. Your overall chance to win is then .230 + .012 + .005 = .247, just short of a take.

In a five-roll position, however, you have an easy take even if not all doubletons play well.

Here, your chance to bear off in four rolls or less is about .42, and your no-cube winning chance is, at least, about .24. Your redoubling equity is substantial, and the position is a definite take.

If the race depends simply on pip count, as in this position, a symmetric position is always a take at any pip count of twelve or more.

Pip counts from eight to eleven are normally drops. There is one exception:

where the slight chance that White will fail to bear off in two rolls gives Black a take. His chance to win is .256.

Conclusion

With one or two checkers remaining, take only if you will have a takeable redouble. With three checkers remaining, take if the pip count is twelve or more. With four checkers remaining, take if your opponent has a substantial chance of missing, usually a gap on a low point.

In a three-roll position, take if there is any substantial chance (say, 6/36) that your opponent will miss. With five or more rolls remaining, always take.

More articles by Bob Floyd

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