Games: Backgammon

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This was the first problem in our Christmas quiz. Should Black redouble, and if he does, should White take or drop?

How does one go about evaluating positions like this? Racing counts are no good and the Thorpe Count is unreliable for such a small number of men on the board. A combination of reference positions and deduction is the method to use. First we notice that Black will be off in three rolls or fewer unless he rolls a number containing a two followed immediately by another two. Double three or better will save him a roll and win the game unless White rolls double six. We know that a pure three-roll position (each side has six men and cannot miss during the bear-off) is a double and a narrow pass. Is White better or worse than a pure three roll position?

If Black does not roll one of his four best doubles then White has six great numbers: 66 - which wins immediately - 55, 44, 33, 65 and 56). These numbers leave Black requiring a double next time. This variation happens 32/36 x 6/36 = 12.3 per cent of the time. If White does not roll one of his best numbers then most of the time he will be left with three or four men against four of Black's men. In a pure two-roll position White will win 14 per cent of the time. Thus he will win this variation 32/36 x 14 = 12.4 per cent. So far we have White winning 24.7 per cent, not quite the 25 per cent required to take.

But he can do no better. We have given White the benefit in our assumptions so far and also ignored the fact that Black may roll a winning double on his second roll. Exact calculation shows that White will lose two points if he drops the redouble but 2.17 points if he takes. Thus the answer is (a) redouble (b) drop.

Christmas quiz results: The copy of Backgammon by Paul Magriel is on its way to Martin and Theresa Hughes from Jersey. Barry McAdam (Ealing) and Graham Titcombe (Kingston-upon-Thames) win Backgammon for Winners by Bill Robertie.