THEORY of MICHAEL'S 432 RULE

- A Small Aid in the Bear Off Situation -

The theory behind

I did a lot of roll outs using GNUBG concerning bear off situation with hits, and from that "Michael's 432 Rule" came up. I've summed up the rule in the following figure:

On the x-axis the number of checkers (or "men") is listed for the player on the bar. Since the player has one checker on the bar the values goes from 0 to 14 checkers.

The y-axis is the calculated probability of winning the game for the player with the closed board, playing to the end with no doubling cube in action. The blue line is when the player with the closed board has an optimal distribution of spare checkers. Typically that is one on the 4-,5- and 6-point. The pink line is when all the spares are placed on the worst position, the ace point.

All probability values are therefore in between the two lines. And between 4 an 9 men left on the acepoint the curves varies linearly, with 10% for each checker. The green area is where the 432-rule can be applied.

The 432 rule is valid within the green area with less than 5% difference from the rollouts.