Black can’t believe the roll he has made! The 3-1 roll does not allow him to bear off even one man!! Black looks at the board and knows at best White is off in 3 more rolls, maybe just 2 more rolls if he should roll a double so Black must now make the best move with the 3-1 roll that gives himself the highest probability of bearing off both men on his next roll.

Black has a choice of three moves. He can move one man from his 5 point to his 1 point; he can move one man from his 6 point to his 2 point; or he can move two men – one from the 6 point to the 3 point and one from the 5 point to the 4 point. The correct move in this case will enable Black to offer White the doubling cube before his next roll. What position does Black require in order to make a reasonable double?