Backgammon Article

Curing Your Short Bearoff Blues
Shortcuts to learning the 2 checker vs. 2 checker bearoff cube actions
by Bob Koca,  July 2007
This is a revised version of an article Bob wrote for
Northern Ohio Backgammon News April 1996.
Thank you to Bob Koca for his kind permission to publish it here.

Correct cube actions for a bearoff position in money play can be classified into 5 categories. Ordered from bad to good for the player on roll they are:
  1. Not good enough to double/beaver if doubled;
  2. Not good enough to double/not a beaver if doubled;
  3. Good enough for an initial double but not a redouble/take if doubled;
  4. Good enough to redouble/take;
  5. Good enough to redouble/pass.
Often a game will come down to one player having two checkers left versus an opponent with one or two checkers left. Since this is common, one can benefit by memorizing a table of these cube actions.

1
2
3
11
12		 4
13		 5 14 6 22 23 24
15		 25 33 34 16 35 26 44 36 45 46 55 56 66
1,2,3,11,12   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
4, 13   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
14   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
22   4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
23   4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5
24, 15   4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5
25   4 4 3 3 3 3 3 4 4 5 5 5 5 5 5 5 5 5 5 5 5
33   1 1 1 1 1 3 3 4 4 5 5 5 5 5 5 5 5 5 5 5 5
34   1 1 1 1 1 3 3 4 4 4 4 5 5 5 5 5 5 5 5 5 5
16   1 1 1 1 1 1 1 3 4 4 4 5 5 5 5 5 5 5 5 5 5
35   1 1 1 1 1 1 1 3 4 4 4 4 5 5 5 5 5 5 5 5 5
26   1 1 1 1 1 1 1 3 4 4 4 4 4 5 5 5 5 5 5 5 5
44   1 1 1 1 1 1 1 1 4 4 4 4 4 4 5 5 5 5 5 5 5
36   1 1 1 1 1 1 1 1 4 4 4 4 4 4 5 5 5 5 5 5 5
45   1 1 1 1 1 1 1 1 4 4 4 4 4 4 5 5 5 5 5 5 5
46   1 1 1 1 1 1 1 1 3 4 4 4 4 4 4 4 4 5 5 5 5
55   1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 5 5 5 5
56   1 1 1 1 1 1 1 1 1 2 2 3 4 4 4 4 4 4 5 5 5
66   1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 4 4 4 4

This table was presented by Danny Kleinman in Vision Laughs at Counting ("The Cub-Off Cube Chart") and by Jeff Ward in The Doubling Cube (Figure 21). Since the table contains 441 entries it would be difficult to remember without some heuristic rules. I believe I have improved on the heuristic rules given by those authors. Once my rules are learned the complete table is known.

Before presenting the rules a couple of preliminary results must be known. First it must be known how many numbers win immediately for a given position. These could easily be counted at the table when the position arises but I think it is worth memorizing. The positions are listed in order of how good they are. Positions with the same number of good rolls are equivalent except for 33 being better than 34, 36 being better than 45 (note how a 2-1 plays better and other rolls are equal), and 55 being better than 56.

Position Number of
Winning Rolls
1 36
2
3
11
12
4 34
13
5 31
14 29
6 27
22 26
23 25
24 23
15
Position Number of
Winning Rolls
25 19
33 17
34 17
16 15
35 14
26 13
44 11
36 10
45 10
46 8
55 6
56 6
66 4

Secondly the Jacoby paradox should be understood. If you have the position 25 then you have an initial double but not a redouble if your opponent has 5, 14, 6, 22 or 23. If your opponent is better than this then you also have a redouble. The paradox is that your opponent's position improving can turn the redouble from incorrect to correct. The explanation is that if your opponent has 5, 14, 6, 22, or 23 then he will have close to 75% win chance if you miss and thus you have given him a powerful cube.







A Jacoby paradox position:
In this position Red should not redouble. But if White's checker is moved to the 4-point (an improvement for White), then Red should redouble.

Except for the Jacoby paradox positions if a position becomes better for the on-turn player (either by making his position better or the opponent's position worse) then the correct action either stays the same or moves down the list. For example suppose you know that 33 vs 23 is an initial double only. Then it can be inferred that 33 vs 24 is at least an initial double. Similarly if the position becomes worse for the on-turn player then the cube action either stays the same or moves up the list. For example, from 33 vs 23 being an initial double only, we know that 34 vs 23 cannot be a redouble.

If the above preliminaries are known then all of the information contained in the chart can be compressed into the following 4 rules:

  1. If you have 27 or more winning numbers then you have a redouble and the opponent has a pass. (The pass is optional if you have exactly 27 good numbers and your opponent has 27 or more).

    For the other positions:

  2. When to take:

    1. If the opponent has the same or fewer winning numbers as you he should pass unless the number of missing rolls are equal and between 19 and 26 or the position is 34 vs 34 or 66 vs 66.
    2. If your opponent has more winning numbers than you then he can take unless the position is (36 or 45) vs 44 or 55 vs 64.

  3. The only positions which are not good enough for an initial double but are not beavers if doubled are 56 vs (33 or 34) and 66 vs 44.

  4. The only initial-double-but-not-a-redouble positions are:

    1. The Jacoby paradox positions,
    2. (33 or 34) vs (22 or 23),
    3. (16 or 35 or 26) vs (24 or 15),
    4. 46 vs 25,
    5. 56 vs 16,
    6. 66 vs (36 or 45).
Do these rules really cover every possible case? Yes, but we might need to think just a little still. A few examples should suffice to show the method. You may want to cover the solution and see if you are able to use the winning numbers list and the rules to determine the answer.

What is the cube action for 44 vs 24?








Red's turn

Answer: Not an initial double/beaver if doubled. Rule 4a says that 26 vs 24 is an initial double only. 44 is worse for you and is not also in that list. Thus it is not an initial double. From rule 3 we know it is a beaver since it is not an initial double and is not one of those 3 positions.

What is the cube action for 56 vs 35?








Red's turn

Answer: Redouble/take. Rule 4e says that 56 vs 16 is an initial double only. Since 35 is worse than 16 it is at least an initial double. 56 vs 35 is not in the initial-double-only list so it is a redouble. Rule 2b says it is a take.

What is the cube action for 55 vs 33?








Red's turn

Answer: Redouble/take. Rule 3 says that 56 vs 33 is not good enough to double. 55 is better than 56 and 55 vs 33 is not in the not-good-enough-to-double list. Thus we conclude it is at least an initial double. Since 55 vs 33 is not in the initial-double-only list we conclude that it is a redouble. Rule 2b says it is a take.

A general description of the method is to first see if the position is exactly covered in the 4 rules listed. If not find a position which is close and is exactly covered in the list of rules and determine if it is better or worse for the play on roll. Adjust the cube action accordingly. What is nice about the list of rules is that a lot of information is obtained from memorizing surprisingly little because the rules contain just the amount one needs to reason everything else out.

Articles by Bob Koca
Articles on cube handling in races
"End Positions in Backgammon," by Edward O. Thorp, 1978.
 
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