Cube Handling in Races 
In a Symmetric Position?



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 oneroll symmetric position is only a take if you have between eighteen and twentyseven rolls that bear off. These are the only oneroll symmetric takes:*


 



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

In a tworoll 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 (55 or 66). 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:
 (large doubletons win)  
+ 
 (your opponent misses, you don't)  
= 
 = .27, a take. 
(Your actual chances are better, because your rolls of 22, 34, 35, 36, 45, 46, and 56 will give you a powerful redouble.)
In such tworoll 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 tworoll 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 A^{2} + B^{2} + C^{2} < 1/2. In Position D, A^{2} + B^{2} + C^{2} = (4 + 225 + 289)/1296 = 518/1296 = .40, well short of 1/2.
Here are some positions that are takes, by the A^{2} + B^{2} + C^{2} < 1/2 criterion:

Position E.
A = 2/36, B = 17/36, C = 17/36. 

Position F.
A = 2/36, B = 19.5/36, C = 14.5/36. 

Position G.
A = 5/36, B = 16.6/36, C = 14.4/36. 
These, however, are clear drops.

A = 2/36, B = 29/36, C = 5/36. A^{2} + B^{2} + C^{2} = 1070/1296, much larger than 1/2, a drop. 

A = 4/36, B = 29/36, C = 3/36. A^{2} + B^{2} + C^{2} = 866/1296, a drop. 

A = 3/36, B = 27/36, C = 6/36. A^{2} + B^{2} + C^{2} = 819/1296, a drop. 
In case you don't want to evaluate A^{2} + B^{2} + C^{2} 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. A^{2} + B^{2} + C^{2} = .504; Black's redoubles give him a take. 

A = 3/36, B = 24/36, C = 9/36. A^{2} + B^{2} + C^{2} = .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.

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


In such positions, where a doubleton gives you a good redouble, you must still have a nocube 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 nocube 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 11 nor 22 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.


In Position J, your chance to go off in three rolls or less is about .36, and your nocube 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 fiveroll position, however, you have an easy take even if not all doubletons play well.

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:

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