Backgammon math

You do not need to be a mathematician to play backgammon well, but a handful of numbers make a real difference. They tell you how risky a blot is, how badly a closed board hurts, and who is ahead in the race. Strong players do not work these out from scratch at the table. They learn a few tables once and use them on every move.

The 36 rolls

Two dice give 36 equally likely outcomes. It helps to imagine one die red and one green. A 6-5 can come as red 6 and green 5 or as red 5 and green 6, so each non-double roll has 2 chances in 36. A double such as 6-6 can come only one way, so it has 1 chance in 36. There are 6 doubles, so a double comes up 6 times in 36, or one roll in six. Every specific non-double, like 3-1, is twice as likely as any specific double.

The totals of the two dice follow the familiar pattern:

In backgammon, though, the total is not the whole story, because doubles are played four times and each die can also be used on its own.

Pip count

The pip count is the total number of points all your checkers must travel to bear off. A checker on your 13-point needs 13 pips, a checker on the bar needs 25. At the start each player has 167 pips: two checkers on the 24-point (48), five on the 13-point (65), three on the 8-point (24) and five on the 6-point (30).

Because doubles count four times, the average roll is worth about 8.17 pips. So a lead of 8 pips is roughly a full roll. Comparing pip counts tells you whether to race or to keep contact, which is one of the main choices discussed in backgammon strategy. A quick way to count is to count by groups, such as five checkers on the 6-point equal 30, rather than one checker at a time.

Shots: how often a blot gets hit

When you leave a blot within reach of an enemy checker, the number of rolls that hit it depends on the distance. Blots up to six points away can be hit with one die (a direct shot). Blots seven or more away need both dice together (an indirect shot).

These counts assume no blocking points in between. Direct shots grow from 11 to 17 as the distance rises, because combinations of both dice add to the rolls that use a single die. Once the distance passes six, the numbers drop sharply. The practical lesson: if you must leave a blot, leaving it seven or more points away is far safer than leaving it close. When two enemy checkers can reach the same blot, you add up the different rolls that hit, counting each roll only once.

Entering from the bar

A checker on the bar enters on an open point in the opponent's home board. The more points your opponent has closed, the harder that becomes:

The rule behind the table is simple: you fail only if both dice show closed numbers. With 4 closed points, each die fails 4 times in 6, so both fail 16 times in 36, and you enter with the other 20. This is why a strong home board is so powerful. Against five closed points you stay out about two times in three.

Putting the numbers to work

Use the shot table when choosing between two plays that both leave a blot: pick the one that leaves fewer hitting rolls, unless the other play gains a lot. Use the entering table to judge how much it costs to be hit. Being hit against a one-point board is a small setback, but against a four- or five-point board it can lose the game. Use the pip count to decide if you are ahead or behind in the race. More practical uses are on the backgammon tips page.