Cuboffs and the Crippled Cube
Danny Kleinman, 1980
Vision Laughs at Counting, Vol. 1, 1980
Once I programmed a computer to calculate all cuboffs — bearoffs where each player has but one or two men left in his home board. In my desk, lie two eight-foot scrolls. One gives all winning percentages (if the games are played to completion); the other gives all equities for the shaker in each of the three possible locations for the cube. There are 729 cuboffs corresponding to the 27 different positions for each player: 21 ways in which two men can be placed on the home board, plus the 6 places on which one man can rest.

These computer outputs are too cumbersome to xerox, too massive to memorize. But they provide interesting data for research. One spin-off from them, for example, is a simple rule for positioning your last two men when you have a choice. The optimum distance between your last two men is 2.7 pips. Approach that as closely as you can. Thus, in order, your best distances are 3, 2, 4, 5, and 0 pips.

The most valuable feature of my cuboff scrolls is the light they shed on cube actions. Cuboffs provide a microcosm of the cube. We may represent them by one or two digits showing the location of the shaker's men, separated by a "v" from one or two digits showing the location of the nonshaker's men. Thus 43v21 is the cuboff where the shaker has men on his 4-point and his 3-point, the nonshaker on his 2-point and his 1-point.

The 43v21 cuboff.

We can classify each cuboff as a nondouble, double, or redouble; and each double as a pass or take. It almost goes without saying that every redouble is also a double and that no nonredoubles can be passes.

Though equities are harder to compute than winning chances, they provide a simple criterion for cube actions. You should double or redouble whenever this gives you higher equity than not doubling. Of course, you should deviate from the theoretically correct actions when your opponent deviates from his best policy in passing or taking. If you are doubled, you should pass when your negative equity reaches beyond the value of the cube, but take otherwise.

Table 1.  List of Cuboff Equities

Equities of the 27 × 27 = 729 cuboff positions for each position of the cube.
Two equities are shown when the player has a proper double. The first equity
assumes he does not double; the second equity assumes he does.

 Pos Prob of Winning Equity by Cube Position (normalized to a 1-cube) Player Opp Player's Cube Centered Cube Opp's Cube
 1 v 1 1 0 +1.00000 +1.00000 +1.00000 1 v 11 1 0 +1.00000 +1.00000 +1.00000 1 v 2 1 0 +1.00000 +1.00000 +1.00000 1 v 21 1 0 +1.00000 +1.00000 +1.00000 1 v 22 1 0 +1.00000 +1.00000 +1.00000 1 v 3 1 0 +1.00000 +1.00000 +1.00000 1 v 31 1 0 +1.00000 +1.00000 +1.00000 1 v 32 1 0 +1.00000 +1.00000 +1.00000 1 v 33 1 0 +1.00000 +1.00000 +1.00000 1 v 4 1 0 +1.00000 +1.00000 +1.00000 1 v 41 1 0 +1.00000 +1.00000 +1.00000 1 v 42 1 0 +1.00000 +1.00000 +1.00000 1 v 43 1 0 +1.00000 +1.00000 +1.00000 1 v 44 1 0 +1.00000 +1.00000 +1.00000 1 v 5 1 0 +1.00000 +1.00000 +1.00000 1 v 51 1 0 +1.00000 +1.00000 +1.00000 1 v 52 1 0 +1.00000 +1.00000 +1.00000 1 v 53 1 0 +1.00000 +1.00000 +1.00000 1 v 54 1 0 +1.00000 +1.00000 +1.00000 1 v 55 1 0 +1.00000 +1.00000 +1.00000 1 v 6 1 0 +1.00000 +1.00000 +1.00000 1 v 61 1 0 +1.00000 +1.00000 +1.00000 1 v 62 1 0 +1.00000 +1.00000 +1.00000 1 v 63 1 0 +1.00000 +1.00000 +1.00000 1 v 64 1 0 +1.00000 +1.00000 +1.00000 1 v 65 1 0 +1.00000 +1.00000 +1.00000 1 v 66 1 0 +1.00000 +1.00000 +1.00000 11 v 1 1 0 +1.00000 +1.00000 +1.00000 11 v 11 1 0 +1.00000 +1.00000 +1.00000 11 v 2 1 0 +1.00000 +1.00000 +1.00000 11 v 21 1 0 +1.00000 +1.00000 +1.00000 11 v 22 1 0 +1.00000 +1.00000 +1.00000 11 v 3 1 0 +1.00000 +1.00000 +1.00000 11 v 31 1 0 +1.00000 +1.00000 +1.00000 11 v 32 1 0 +1.00000 +1.00000 +1.00000 11 v 33 1 0 +1.00000 +1.00000 +1.00000 11 v 4 1 0 +1.00000 +1.00000 +1.00000 11 v 41 1 0 +1.00000 +1.00000 +1.00000 11 v 42 1 0 +1.00000 +1.00000 +1.00000 11 v 43 1 0 +1.00000 +1.00000 +1.00000 11 v 44 1 0 +1.00000 +1.00000 +1.00000 11 v 5 1 0 +1.00000 +1.00000 +1.00000 11 v 51 1 0 +1.00000 +1.00000 +1.00000 11 v 52 1 0 +1.00000 +1.00000 +1.00000 11 v 53 1 0 +1.00000 +1.00000 +1.00000 11 v 54 1 0 +1.00000 +1.00000 +1.00000 11 v 55 1 0 +1.00000 +1.00000 +1.00000 11 v 6 1 0 +1.00000 +1.00000 +1.00000 11 v 61 1 0 +1.00000 +1.00000 +1.00000 11 v 62 1 0 +1.00000 +1.00000 +1.00000 11 v 63 1 0 +1.00000 +1.00000 +1.00000 11 v 64 1 0 +1.00000 +1.00000 +1.00000 11 v 65 1 0 +1.00000 +1.00000 +1.00000 11 v 66 1 0 +1.00000 +1.00000 +1.00000 2 v 1 1 0 +1.00000 +1.00000 +1.00000 2 v 11 1 0 +1.00000 +1.00000 +1.00000 2 v 2 1 0 +1.00000 +1.00000 +1.00000 2 v 21 1 0 +1.00000 +1.00000 +1.00000 2 v 22 1 0 +1.00000 +1.00000 +1.00000 2 v 3 1 0 +1.00000 +1.00000 +1.00000 2 v 31 1 0 +1.00000 +1.00000 +1.00000 2 v 32 1 0 +1.00000 +1.00000 +1.00000 2 v 33 1 0 +1.00000 +1.00000 +1.00000 2 v 4 1 0 +1.00000 +1.00000 +1.00000 2 v 41 1 0 +1.00000 +1.00000 +1.00000 2 v 42 1 0 +1.00000 +1.00000 +1.00000 2 v 43 1 0 +1.00000 +1.00000 +1.00000 2 v 44 1 0 +1.00000 +1.00000 +1.00000 2 v 5 1 0 +1.00000 +1.00000 +1.00000 2 v 51 1 0 +1.00000 +1.00000 +1.00000 2 v 52 1 0 +1.00000 +1.00000 +1.00000 2 v 53 1 0 +1.00000 +1.00000 +1.00000 2 v 54 1 0 +1.00000 +1.00000 +1.00000 2 v 55 1 0 +1.00000 +1.00000 +1.00000 2 v 6 1 0 +1.00000 +1.00000 +1.00000 2 v 61 1 0 +1.00000 +1.00000 +1.00000 2 v 62 1 0 +1.00000 +1.00000 +1.00000 2 v 63 1 0 +1.00000 +1.00000 +1.00000 2 v 64 1 0 +1.00000 +1.00000 +1.00000 2 v 65 1 0 +1.00000 +1.00000 +1.00000 2 v 66 1 0 +1.00000 +1.00000 +1.00000 21 v 1 1 0 +1.00000 +1.00000 +1.00000 21 v 11 1 0 +1.00000 +1.00000 +1.00000 21 v 2 1 0 +1.00000 +1.00000 +1.00000 21 v 21 1 0 +1.00000 +1.00000 +1.00000 21 v 22 1 0 +1.00000 +1.00000 +1.00000 21 v 3 1 0 +1.00000 +1.00000 +1.00000 21 v 31 1 0 +1.00000 +1.00000 +1.00000 21 v 32 1 0 +1.00000 +1.00000 +1.00000 21 v 33 1 0 +1.00000 +1.00000 +1.00000 21 v 4 1 0 +1.00000 +1.00000 +1.00000 21 v 41 1 0 +1.00000 +1.00000 +1.00000 21 v 42 1 0 +1.00000 +1.00000 +1.00000 21 v 43 1 0 +1.00000 +1.00000 +1.00000 21 v 44 1 0 +1.00000 +1.00000 +1.00000 21 v 5 1 0 +1.00000 +1.00000 +1.00000 21 v 51 1 0 +1.00000 +1.00000 +1.00000 21 v 52 1 0 +1.00000 +1.00000 +1.00000 21 v 53 1 0 +1.00000 +1.00000 +1.00000 21 v 54 1 0 +1.00000 +1.00000 +1.00000 21 v 55 1 0 +1.00000 +1.00000 +1.00000 21 v 6 1 0 +1.00000 +1.00000 +1.00000 21 v 61 1 0 +1.00000 +1.00000 +1.00000 21 v 62 1 0 +1.00000 +1.00000 +1.00000 21 v 63 1 0 +1.00000 +1.00000 +1.00000 21 v 64 1 0 +1.00000 +1.00000 +1.00000 21 v 65 1 0 +1.00000 +1.00000 +1.00000 21 v 66 1 0 +1.00000 +1.00000 +1.00000 22 v 1 0.722 0.278 +0.44444 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 11 0.722 0.278 +0.44444 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 2 0.722 0.278 +0.44444 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 21 0.722 0.278 +0.44444 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 22 0.799 0.201 +0.59877 d/t +0.95062 +0.47531 d/t +0.95062 +0.47531 22 v 3 0.722 0.278 +0.44444 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 31 0.738 0.262 +0.47531 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 32 0.807 0.193 +0.61420 d/p +1.00000 +0.50617 d/p +1.00000 +0.50617 22 v 33 0.869 0.131 +0.73765 d/p +1.00000 +0.73765 d/p +1.00000 +0.73765 22 v 4 0.738 0.262 +0.47531 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 41 0.776 0.224 +0.55247 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 42 0.823 0.177 +0.64506 d/p +1.00000 +0.56790 d/p +1.00000 +0.56790 22 v 43 0.869 0.131 +0.73765 d/p +1.00000 +0.73765 d/p +1.00000 +0.73765 22 v 44 0.915 0.085 +0.83025 d/p +1.00000 +0.83025 d/p +1.00000 +0.83025 22 v 5 0.761 0.239 +0.52160 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 51 0.823 0.177 +0.64506 d/p +1.00000 +0.56790 d/p +1.00000 +0.56790 22 v 52 0.853 0.147 +0.70679 d/p +1.00000 +0.69136 d/p +1.00000 +0.69136 22 v 53 0.892 0.108 +0.78395 d/p +1.00000 +0.78395 d/p +1.00000 +0.78395 22 v 54 0.923 0.077 +0.84568 d/p +1.00000 +0.84568 d/p +1.00000 +0.84568 22 v 55 0.954 0.046 +0.90741 d/p +1.00000 +0.90741 d/p +1.00000 +0.90741 22 v 6 0.792 0.208 +0.58333 d/t +0.88889 +0.44444 d/t +0.88889 +0.44444 22 v 61 0.884 0.116 +0.76852 d/p +1.00000 +0.76852 d/p +1.00000 +0.76852 22 v 62 0.9 0.1 +0.79938 d/p +1.00000 +0.79938 d/p +1.00000 +0.79938 22 v 63 0.923 0.077 +0.84568 d/p +1.00000 +0.84568 d/p +1.00000 +0.84568 22 v 64 0.938 0.062 +0.87654 d/p +1.00000 +0.87654 d/p +1.00000 +0.87654 22 v 65 0.954 0.046 +0.90741 d/p +1.00000 +0.90741 d/p +1.00000 +0.90741 22 v 66 0.969 0.031 +0.93827 d/p +1.00000 +0.93827 d/p +1.00000 +0.93827 3 v 1 1 0 +1.00000 +1.00000 +1.00000 3 v 11 1 0 +1.00000 +1.00000 +1.00000 3 v 2 1 0 +1.00000 +1.00000 +1.00000 3 v 21 1 0 +1.00000 +1.00000 +1.00000 3 v 22 1 0 +1.00000 +1.00000 +1.00000 3 v 3 1 0 +1.00000 +1.00000 +1.00000 3 v 31 1 0 +1.00000 +1.00000 +1.00000 3 v 32 1 0 +1.00000 +1.00000 +1.00000 3 v 33 1 0 +1.00000 +1.00000 +1.00000 3 v 4 1 0 +1.00000 +1.00000 +1.00000 3 v 41 1 0 +1.00000 +1.00000 +1.00000 3 v 42 1 0 +1.00000 +1.00000 +1.00000 3 v 43 1 0 +1.00000 +1.00000 +1.00000 3 v 44 1 0 +1.00000 +1.00000 +1.00000 3 v 5 1 0 +1.00000 +1.00000 +1.00000 3 v 51 1 0 +1.00000 +1.00000 +1.00000 3 v 52 1 0 +1.00000 +1.00000 +1.00000 3 v 53 1 0 +1.00000 +1.00000 +1.00000 3 v 54 1 0 +1.00000 +1.00000 +1.00000 3 v 55 1 0 +1.00000 +1.00000 +1.00000 3 v 6 1 0 +1.00000 +1.00000 +1.00000 3 v 61 1 0 +1.00000 +1.00000 +1.00000 3 v 62 1 0 +1.00000 +1.00000 +1.00000 3 v 63 1 0 +1.00000 +1.00000 +1.00000 3 v 64 1 0 +1.00000 +1.00000 +1.00000 3 v 65 1 0 +1.00000 +1.00000 +1.00000 3 v 66 1 0 +1.00000 +1.00000 +1.00000 31 v 1 0.944 0.056 +0.88889 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 11 0.944 0.056 +0.88889 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 2 0.944 0.056 +0.88889 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 21 0.944 0.056 +0.88889 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 22 0.96 0.04 +0.91975 d/p +1.00000 +0.89506 d/p +1.00000 +0.89506 31 v 3 0.944 0.056 +0.88889 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 31 0.948 0.052 +0.89506 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 32 0.961 0.039 +0.92284 d/p +1.00000 +0.90123 d/p +1.00000 +0.90123 31 v 33 0.974 0.026 +0.94753 d/p +1.00000 +0.94753 d/p +1.00000 +0.94753 31 v 4 0.948 0.052 +0.89506 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 41 0.955 0.045 +0.91049 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 42 0.965 0.035 +0.92901 d/p +1.00000 +0.91358 d/p +1.00000 +0.91358 31 v 43 0.974 0.026 +0.94753 d/p +1.00000 +0.94753 d/p +1.00000 +0.94753 31 v 44 0.983 0.017 +0.96605 d/p +1.00000 +0.96605 d/p +1.00000 +0.96605 31 v 5 0.952 0.048 +0.90432 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 51 0.965 0.035 +0.92901 d/p +1.00000 +0.91358 d/p +1.00000 +0.91358 31 v 52 0.971 0.029 +0.94136 d/p +1.00000 +0.93827 d/p +1.00000 +0.93827 31 v 53 0.978 0.022 +0.95679 d/p +1.00000 +0.95679 d/p +1.00000 +0.95679 31 v 54 0.985 0.015 +0.96914 d/p +1.00000 +0.96914 d/p +1.00000 +0.96914 31 v 55 0.991 0.009 +0.98148 d/p +1.00000 +0.98148 d/p +1.00000 +0.98148 31 v 6 0.958 0.042 +0.91667 d/p +1.00000 +0.88889 d/p +1.00000 +0.88889 31 v 61 0.977 0.023 +0.95370 d/p +1.00000 +0.95370 d/p +1.00000 +0.95370 31 v 62 0.98 0.02 +0.95988 d/p +1.00000 +0.95988 d/p +1.00000 +0.95988 31 v 63 0.985 0.015 +0.96914 d/p +1.00000 +0.96914 d/p +1.00000 +0.96914 31 v 64 0.988 0.012 +0.97531 d/p +1.00000 +0.97531 d/p +1.00000 +0.97531 31 v 65 0.991 0.009 +0.98148 d/p +1.00000 +0.98148 d/p +1.00000 +0.98148 31 v 66 0.994 0.006 +0.98765 d/p +1.00000 +0.98765 d/p +1.00000 +0.98765 32 v 1 0.694 0.306 +0.38889 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 11 0.694 0.306 +0.38889 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 2 0.694 0.306 +0.38889 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 21 0.694 0.306 +0.38889 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 22 0.779 0.221 +0.55864 d/t +0.84568 +0.42284 d/t +0.84568 +0.42284 32 v 3 0.694 0.306 +0.38889 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 31 0.711 0.289 +0.42284 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 32 0.788 0.212 +0.57562 d/t +0.91358 +0.45679 d/t +0.91358 +0.45679 32 v 33 0.856 0.144 +0.71142 d/p +1.00000 +0.71142 d/p +1.00000 +0.71142 32 v 4 0.711 0.289 +0.42284 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 41 0.754 0.246 +0.50772 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 42 0.805 0.195 +0.60957 d/p +1.00000 +0.52469 d/p +1.00000 +0.52469 32 v 43 0.856 0.144 +0.71142 d/p +1.00000 +0.71142 d/p +1.00000 +0.71142 32 v 44 0.907 0.093 +0.81327 d/p +1.00000 +0.81327 d/p +1.00000 +0.81327 32 v 5 0.737 0.263 +0.47377 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 51 0.805 0.195 +0.60957 d/p +1.00000 +0.52469 d/p +1.00000 +0.52469 32 v 52 0.839 0.161 +0.67747 d/p +1.00000 +0.66049 d/p +1.00000 +0.66049 32 v 53 0.881 0.119 +0.76235 d/p +1.00000 +0.76235 d/p +1.00000 +0.76235 32 v 54 0.915 0.085 +0.83025 d/p +1.00000 +0.83025 d/p +1.00000 +0.83025 32 v 55 0.949 0.051 +0.89815 d/p +1.00000 +0.89815 d/p +1.00000 +0.89815 32 v 6 0.771 0.229 +0.54167 d/t +0.77778 +0.38889 d/t +0.77778 +0.38889 32 v 61 0.873 0.127 +0.74537 d/p +1.00000 +0.74537 d/p +1.00000 +0.74537 32 v 62 0.89 0.11 +0.77932 d/p +1.00000 +0.77932 d/p +1.00000 +0.77932 32 v 63 0.915 0.085 +0.83025 d/p +1.00000 +0.83025 d/p +1.00000 +0.83025 32 v 64 0.932 0.068 +0.86420 d/p +1.00000 +0.86420 d/p +1.00000 +0.86420 32 v 65 0.949 0.051 +0.89815 d/p +1.00000 +0.89815 d/p +1.00000 +0.89815 32 v 66 0.966 0.034 +0.93210 d/p +1.00000 +0.93210 d/p +1.00000 +0.93210 33 v 1 0.472 0.528 −0.05556 −0.05556 −0.05556 33 v 11 0.472 0.528 −0.05556 −0.05556 −0.05556 33 v 2 0.472 0.528 −0.05556 −0.05556 −0.05556 33 v 21 0.472 0.528 −0.05556 −0.05556 −0.05556 33 v 22 0.619 0.381 +0.23765 +0.00309 d/t +0.00617 +0.00309 33 v 3 0.472 0.528 −0.05556 −0.05556 −0.05556 33 v 31 0.502 0.498 +0.00309 −0.05556 −0.05556 33 v 32 0.633 0.367 +0.26698 +0.06173 d/t +0.12346 +0.06173 33 v 33 0.751 0.249 +0.50154 d/p +1.00000 +0.50154 d/p +1.00000 +0.50154 33 v 4 0.502 0.498 +0.00309 −0.05556 −0.05556 33 v 41 0.575 0.425 +0.14969 −0.05556 −0.05556 33 v 42 0.663 0.337 +0.32562 d/t +0.35802 +0.17901 d/t +0.35802 +0.17901 33 v 43 0.751 0.249 +0.50154 d/p +1.00000 +0.50154 d/p +1.00000 +0.50154 33 v 44 0.839 0.161 +0.67747 d/p +1.00000 +0.67747 d/p +1.00000 +0.67747 33 v 5 0.546 0.454 +0.09105 −0.05556 −0.05556 33 v 51 0.663 0.337 +0.32562 d/t +0.35802 +0.17901 d/t +0.35802 +0.17901 33 v 52 0.721 0.279 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−0.55556 −0.55556 64 v 51 0.477 0.523 −0.01166 −0.23457 −0.23457 64 v 52 0.556 0.444 +0.15569 +0.08916 d/t +0.12560 +0.06280 64 v 53 0.655 0.345 +0.36512 d/t +0.58375 +0.34752 d/t +0.58375 +0.29188 64 v 54 0.735 0.265 +0.53388 d/t +0.91984 +0.52455 d/t +0.91984 +0.45992 64 v 55 0.816 0.184 +0.70459 d/p +1.00000 +0.70412 d/p +1.00000 +0.62734 64 v 6 0.399 0.601 −0.17901 −0.55556 −0.55556 64 v 61 0.634 0.366 +0.32305 d/t +0.50077 +0.30350 d/t +0.50077 +0.25039 64 v 62 0.674 0.326 +0.40695 d/t +0.66580 +0.39107 d/t +0.66580 +0.33290 64 v 63 0.734 0.266 +0.53314 d/t +0.91464 +0.52308 d/t +0.91464 +0.45732 64 v 64 0.776 0.224 +0.61997 d/p +1.00000 +0.61648 d/p +1.00000 +0.54624 64 v 65 0.82 0.18 +0.70780 d/p +1.00000 +0.70622 d/p +1.00000 +0.63132 64 v 66 0.865 0.135 +0.79792 d/p +1.00000 +0.79611 d/p +1.00000 +0.72125 65 v 1 0.167 0.833 −0.66667 −0.66667 −0.66667 65 v 11 0.167 0.833 −0.66667 −0.66667 −0.66667 65 v 2 0.167 0.833 −0.66667 −0.66667 −0.66667 65 v 21 0.167 0.833 −0.66667 −0.66667 −0.66667 65 v 22 0.363 0.637 −0.24314 −0.58951 −0.58951 65 v 3 0.167 0.833 −0.66667 −0.66667 −0.66667 65 v 31 0.206 0.794 −0.58196 −0.66667 −0.66667 65 v 32 0.383 0.617 −0.20079 −0.51235 −0.51235 65 v 33 0.54 0.46 +0.13803 +0.08076 +0.02538 65 v 4 0.206 0.794 −0.58196 −0.66667 −0.66667 65 v 41 0.304 0.696 −0.37020 −0.66667 −0.66667 65 v 42 0.422 0.578 −0.11608 −0.35802 −0.35802 65 v 43 0.54 0.46 +0.13824 +0.08076 +0.02538 65 v 44 0.66 0.34 +0.39379 d/t +0.55595 +0.35085 d/t +0.55595 +0.27797 65 v 5 0.265 0.735 −0.45490 −0.66667 −0.66667 65 v 51 0.422 0.578 −0.11608 −0.35802 −0.35802 65 v 52 0.501 0.499 +0.05354 −0.04527 −0.07373 65 v 53 0.601 0.399 +0.26628 d/t +0.29701 +0.21263 d/t +0.29701 +0.14851 65 v 54 0.682 0.318 +0.43953 d/t +0.65854 +0.40376 d/t +0.65854 +0.32927 65 v 55 0.766 0.234 +0.61777 d/p +1.00000 +0.60389 d/p +1.00000 +0.51371 65 v 6 0.344 0.656 −0.28549 −0.66667 −0.66667 65 v 61 0.58 0.42 +0.22295 +0.16821 d/t +0.21399 +0.10700 65 v 62 0.62 0.38 +0.30863 d/t +0.38130 +0.25769 d/t +0.38130 +0.19065 65 v 63 0.682 0.318 +0.43801 d/t +0.64984 +0.40071 d/t +0.64984 +0.32492 65 v 64 0.726 0.274 +0.53102 d/t +0.85607 +0.50957 d/t +0.85607 +0.42804 65 v 65 0.772 0.228 +0.62752 d/p +1.00000 +0.61690 d/p +1.00000 +0.52867 65 v 66 0.823 0.177 +0.73071 d/p +1.00000 +0.72358 d/p +1.00000 +0.62902 66 v 1 0.111 0.889 −0.77778 −0.77778 −0.77778 66 v 11 0.111 0.889 −0.77778 −0.77778 −0.77778 66 v 2 0.111 0.889 −0.77778 −0.77778 −0.77778 66 v 21 0.111 0.889 −0.77778 −0.77778 −0.77778 66 v 22 0.298 0.702 −0.36797 −0.71296 −0.71296 66 v 3 0.111 0.889 −0.77778 −0.77778 −0.77778 66 v 31 0.148 0.852 −0.69582 −0.77778 −0.77778 66 v 32 0.316 0.684 −0.32699 −0.64335 −0.64335 66 v 33 0.465 0.535 +0.00086 −0.10374 −0.16872 66 v 4 0.148 0.852 −0.69582 −0.77778 −0.77778 66 v 41 0.242 0.758 −0.49091 −0.77778 −0.77778 66 v 42 0.354 0.646 −0.24503 −0.50274 −0.50274 66 v 43 0.466 0.534 +0.00160 −0.10355 −0.16853 66 v 44 0.581 0.419 +0.25085 +0.16092 +0.07090 66 v 5 0.204 0.796 −0.57287 −0.77778 −0.77778 66 v 51 0.354 0.646 −0.24503 −0.50274 −0.50274 66 v 52 0.429 0.571 −0.08036 −0.21879 −0.25146 66 v 53 0.524 0.476 +0.12701 +0.02592 −0.05330 66 v 54 0.604 0.396 +0.29926 +0.21659 d/t +0.24807 +0.12403 66 v 55 0.688 0.312 +0.48088 d/t +0.64699 +0.43194 d/t +0.64699 +0.32349 66 v 6 0.279 0.721 −0.40895 −0.77778 −0.77778 66 v 61 0.503 0.497 +0.08356 −0.01713 −0.09275 66 v 62 0.543 0.457 +0.16799 +0.06942 −0.01340 66 v 63 0.603 0.397 +0.29625 +0.21224 d/t +0.23724 +0.11862 66 v 64 0.649 0.351 +0.39456 d/t +0.46463 +0.33199 d/t +0.46463 +0.23232 66 v 65 0.698 0.302 +0.50072 d/t +0.71459 +0.46564 d/t +0.71459 +0.35730 66 v 66 0.754 0.246 +0.62174 d/t +0.96126 +0.60312 d/t +0.96126 +0.48063

It would be convenient if we could relate cube actions to the easier-to-compute winning chances. But the value of the cube to its owner precludes any simple relationship between cube actions and winning chances. Nonetheless, there are two models suggested by different backgammon theorists: the Classical and the Continuous.

The Classical Criterion tells you to take only with at least 25% winning chances. The Continuous Criterion tells you to pass only with at most 20% chances. Each expresses a half-truth. The Classical Model ignores the value of the cube to the owner. The Continuous Model assumes that the cube can be used with perfect efficiency — that a game always turns around gradually, allowing the owner of the cube to redouble when his winning chances reach exactly 80%, when his opponent has an equally unpleasant pass or take.

But we may combine the true half of both these theories to form a partial strategy: You may always take with at least 25% chances, and you may always pass with at most 20% chances. Between these two limits you must consider the value of owning the cube. The cube may be dead or alive or more or less crippled. The more Crippled the cube, the more your chances must exceed 20% in order to warrant a take.

One way for the cube to be dead is for this to be the last shake of the game. If the shaker misses, he loses; there is no need for the nonshaker to return the cube. The 52v11 cuboff illustrates a dead cube, for all 36 numbers get the nonshaker off. On the last shake, even this slim advantage, with winning chances under 53%, justifies a redouble. Another dead cube occurs in 6v21. This is an optional pass despite the 25% winning chances.

A second way for the cube to be dead occurs when the game can never turn around sufficiently for the nonshaker to redouble. We may think of this as a two-shake bearoff, where the nonshaker has fewer than 19 winning numbers. In 33v33, the nonshaker will win 323 games in 1296, but in none of them will he become the favorite. The cube is thus worthless to him, and he should pass despite his 24.923% winning chances, just a shade under 25%

With black to roll, white never gets a chance to make a proper redouble.

At the extremes of 18 (or fewer) and 36 winning numbers for the nonshaker, the cube is totally dead. Exactly halfway between, at 27 winning numbers, the cube is totally alive. Between 19 and 26, or between 28 and 35, the cube is crippled to varying degrees — the closer to 27, the less crippled.

The 6v6 cuboff shows a totally alive cube. With 27 winning numbers, the nonshaker has an optional take, even with only 18.75% chances. The 22v22 cuboff demonstrates a cube crippled only to the tiniest degree. At 20.062%, it is a mandatory take. The nonshaker has 26 winning numbers if the shaker misses.

My scrolls of cuboff equities and winning chances reveal that prior to the last shake, better than 55% chances are always required to double or redouble. With at least 61% winning chances, a double is justified in any cuboff; and with at least 68% winning chances, a redouble. But what about doubles in the 55% to 60% range, and redoubles in the 55% to 68% range?

Once again, we may use the concept of the crippled cube! 52v32 offers better than 67% chances, yet it is not a redouble. The nonshaker has 25 winning numbers, quite close to the perfect 27; the cube is scarcely crippled at all. In contrast, the 52v31 cuboff is a fine redouble despite chances just above 55%. With 34 winning numbers for the nonshaker, the cube is crippled severely, almost dead.

In The Backgammon Book, Jacoby and Crawford explain the Jacoby Paradox as follows. 52v51 is a redouble, but the poorer cuboff 52v41 is not. "The reason is that if you keep the cube on your side and fail to get both your men off, you may get a second turn."

I find it more paradoxical that 52v31 is a redouble while 52v32 is not. Redoubling in 52v31 risks depriving you of a second turn, while redoubling in 52v32 cannot (for you will have a take if you miss). The concept of the crippled cube resolves the paradox of this 55% redouble and 67% nonredouble, and shows the usual explanation of the Jacoby Paradox to be mistaken.

Another sound redouble with relatively low (56%) chances occurs in 55v33. The cube is very crippled, and a miss seldom fetches the return of the cube, for the nonshaker seldom becomes more than a small favorite.

In doubling from the center, the concept of crippling still applies, but over a narrower range of winning chances. In 33v6, your winning chances actually exceed 60% — if the game is played to completion. But it won't be if you miss; you will be doubled out with perfect efficiency since your opponent has 27 winning numbers. You are really the underdog, and you must not double. On the other hand, in 64v52 you are less than a 56% favorite. But your opponent has only 19 winning numbers if you miss, crippling the cube severely. Only a roll of 2-1 will fetch a return of the cube, so you should double now.

Cuboffs of only two shakes lend themselves to exact equity calculations fairly readily, even where further cube turns are possible. Suppose, for example you get doubled in 32v51. If you take, you will lose immediately 25 times in 36, or 900 times in a cross-section of 1296 games; all with the cube at 2. But the remaining 11 times out of 36 you will redouble. Since you have only 23 winning numbers, your opponent will take the cube at 4. Your net in each of these 11 times will be 10 wins in 36 games (23 − 13). This nets to  440 − 1800,  or  −1360  points in 1296 games. You should pass, thus losing only 1296 points in 1296 games.

You can simplify your calculations in cuboffs to exclude equities and cube turns by using a technique I call excess eighteen. The excess of winning numbers over 18 measures the liveliness of the cube. Add this excess to your count of winning numbers (making sure, of course, that you don't count more than 36). Then you can compute "wins in 1296" pure and simple, noting that 324 wins divides passes from takes.

Let's recompute 32v51 using excess eighteen. The excess of your 23 winning numbers over 18 is 5. So add 5 to your winning numbers, counting them as 28, Your opponent misses with 11 numbers, so you may figure that you will win 308 (11 × 28) games in 1296. You don't need to adjust the Classical 25% Criterion of 324 wins in 1296 to account for the worth of the cube ownership. Excess eighteen has already done that for you, automatically, and you should pass.

How can the analysis of cuboffs be extended to help us with other cube actions? Let us note similarities and differences between the cube in cuboffs and in midgame positions.

With very few shakes left, cuboffs tend to be very volatile, since any single shake can make a tremendous difference in the game. Cuboffs are very discontinuous also, with frequent highly crippled cubes. In these crippled-cube cuboffs, the conservative passes and liberal doubles of the Classical Model are often justified. But most ordinary backgammon positions approximate the Continuous Model more closely. In those games, you should let the worth of owning the cube induce you to take with chances only slightly better than 20%. And you should be wary of doubling before your chances threaten to rise above 80% by your next turn.

You should be on the lookout for very volatile, discontinuous games, however, where you can apply the lessons of the cuboff. Certain positions can change abruptly. On the next shake you may either hit a shot or convert to a pure race; or you can free your backward man or see your board crumble. These resemble cuboffs.

Let us consider just one typical situation. Owning the cube, you play a holding game. Your opponent bears in awkwardly, until a very bad shake forces him to leave you many winning numbers. Perhaps a direct double shot with some extra combinations that hit. If you hit, your opponent will easily pass your redouble at the next turn. Should you redouble him now, while he will still take?

The concept of the crippled cube leads us to the right answer. What if you miss your shot? Will the cube be dead for your opponent or alive? There are the same two ways for the cube to be crippled — your miss can leave your opponent still only a slight favorite to win the game, or it can give him a virtual lock. Either way, you may redouble, just as you do in 55v35, 52v11, and 52v31 cuboffs with the cube crippled.

But when the cube is least crippled, you must hold on to it for dear life. When your chances may range from 16% to 33% after you miss your shot, you can't afford to redouble and find the cube coming back at you. Your opponent's chances if you miss will be close to 75%, with the cube very alive for him. Save your redouble for your last double shot in an otherwise lost position.

In most backgammon positions, you can think of the worth of owning the cube as proportional to the winning chances of the cube owner. By owning the cube, in effect, you multiply your winning chances by 125%. That is why you can take with only 20% chances in a game with a long way to go. Owning the cube multiplies the 20% by 125%, giving you 25% chances in effect.

As your position improves, the worth of the cube to you increases proportionally and gradually, at one quarter the rate of your winning chances. When your winning chances reach 80%, the cube adds another 20%, giving you 100% chances in effect, a double out.

But beyond 80%, the worth of owning the cube shrinks instead of rising, and exactly by the amount by which your winning chances increase. This shrinking of the worth of the cube completely cancels out the improvement in your position. That is why it is such a terrible sin to wait too long before doubling or redoubling, unless you are playing a very liberal opponent who will still take even with much poorer than 20% winning chances.