OK, so have you worked out the pip counts? Check the positions below to see if you are correct and for the description for the shifting method uesd.

Click on the postion you'd like to view:

Position 14 - Position 15 - Position 16 - Position 17

Conclusion - Back to Cluster Counting

By using Two Way Mental Shifting the position becomes:

Position 14

Position 14a

Black = 100   White = 84
Black's 100 pips can be counted in three clusters: 40 (5-Prime from the 6-point to the 2-point) + 50 (Mirrors on the 7-point and the 18-point) + 10. White's 84 pips can be counted in two clusters: 44 (5-Prime + 4) + 40 (four 10s).

Position 15

Black = ?  White = ?

After shifting, Black's and White's positions becomes:

Position 15a

Position 15b

Black = 157    White = 149

Black's position can be counted in three clusters: 30 (six 5s) + 43 (RP#5 - five 8s + 3) + 84 (four 20s + 4). White's pips can be counted in three clusters: 42 (eight 5s + 2) + 40 (RP#7) + 67 (three 20s +7).

After shifting, Black's and White's positions becomes:

Position 16

Position 16a

Black = 106  White = 100

Black's pips can be counted in two clusters: 66 (twelve 5s + 6) + 40 (two 20s). White's 100 pips can be counted in two clusters: 30 (six 5s) + 70 (RP#4 again + 10 for two checkers moved from the 13-point to the 8-point).

Note that in Position 16a White has only 14 checkers. The two checkers originally on the 3-point were shifted in different directions - one checker to the 6-point and the other checker off the board.

As previously noted, with Cluster Counting, there is almost always more than one correct way to count a position. You should use whichever cluster formations you can quickly visualize. For example, look at Position 17. With a minimum of shifting, Black's pip count can be quickly counted in several different ways:
Position 17
a. 63 (5-Prime +3) + 75 (five 13s + 10 by shifting two checkers from the 18-point to the 13-point);

b.63 (5-Prime + 3) + 62 (RP#6) + 13 (spare checker on the 13-point);

c.50 (Mirrors on the 12- and 13-points) + 50 (Mirrors on the 7- and 18-points) + 30 (six 5s) + 8 (Checker on the 8-point).

Black = 138


Well, that's the system. Certainly my list of seven Reference Positions is by no means inclusive. You probably already know or will discover other positions that can be added to the list.

Will mastering the Cluster Counting technique improve your game, or at least make one tedious aspect of backgammon more enjoyable? Count on it.


This article was previously published in the Chicago Point (visit their web site!), Issue #52, November 1992. It was made available on the Internet in 1997 by Kate McCollough with the author's permission.

Thanks to Jack and to Bill Davis of the Chicago Point. Thanks also to Kevin Bastian for creating the graphics for the original page. And thanks to Kate McCollough for creating the original HTML version of the article and putting it up on the web HERE via Tom Keith's Backgammon Galore.

Originally created: March 31, 1997. Revamped by Michael Crane 27 July 2007

Back to Cluster Counting