During the bearoff, the ownership of the doubling cube is of some importance. In the ten positions below Black is on the roll and owns the doubling cube. No gammons are possible. Only Black owning the cube can cash the game with 75% winning chances; White must finish the entire bearoff if Black chooses to wait. Each of the ten positions is a reference redouble, meaning to within a few milli-points of equity (less than 10mp) Black has the same positive equity in the position whether or not Black recubes now. Of course, White will correctly take the cube.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 1mp (one milli-point of equity). If the doubling cube was centered instead, Black would certainly double White and then White would still take (not a reference first cube but 48mp into the doubling window as a first cube). An estimate of the cube worth is about forty-seven cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 1mp (one milli-point of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 33mp into the doubling window as a first cube). An estimate of the cube worth is about thirty-two cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is ND (no double) by 2mp (two milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 30mp into the doubling window as a first cube). An estimate of the cube worth is about thirty-two cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 2mp (two milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 40mp into the doubling window as a first cube). An estimate of the cube worth is about thirty-eight cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is ND by 9mp (nine milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 31mp into the doubling window as a first cube). An estimate of the cube worth is about forty cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 7mp (seven milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 49mp into the doubling window as a first cube). An estimate of the cube worth is about forty-two cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is ND by 8mp (eight milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 35mp into the doubling window as a first cube). An estimate of the cube worth is about forty-three cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is ND by 1mp (one milli-point of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 34mp into the doubling window as a first cube). An estimate of the cube worth is about thirty-five cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 2mp (two milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 35mp into the doubling window as a first cube). An estimate of the cube worth is about thirty-three cents in ten dollars.

Black on the roll. Black can either recube to 4 or Black can wait and play on at a stake of 2. The position is RD/T by 9mp (nine milli-points of equity). If the doubling cube was centered instead, Black would double White and then White would still take (not a reference first cube but 25mp into the doubling window as a first cube). An estimate of the cube worth is about sixteen cents in ten dollars.