
Cluster Counting

Jack Kissane November 1992

This article originally appeared in the November 1992 issue of Chicago Point.
Thank you to Jack Kissane and Bill Davis for their kind permission to reproduce it here.


Introduction


Jack Kissane, backgammon master from Albany, New York, is known in many
chouette circles as the fastest pip counter in the world. In a June 1989 Chicago Point interview, Kissane claimed
that he can count almost any backgammon position within five seconds.
For the first time anywhere, Jack Kissane shares his counting techniques with
the backgammon community. Enjoy! 
Pip counting. How do you view it? An annoyance? A necessity? Just part of the
game? Some backgammon players can't or won't be bothered doing a pip count.
Others use the count as a crutch, basing far too many checker moves on it.
After
a hard day of match play or during an allnight chouette, pip counting can be
sheer torture, draining our limited supply of "thinking" energy. However, once
or twice a game, knowing the count is critical for making the right checker play
or, more importantly, the correct cube decision.
Over the years, I have developed a system of pip counting that significantly
reduces the amount of time needed to count a position. I call it Cluster
Counting. Hopefully, this fairly simple system will help you minimize the
drudgery of pip counting and thus increase your enjoyment of the game.
Basically, Cluster Counting involves the mental shifting of checkers to form
patterns of reference positions (RP's) whose pip totals end in zero (with
two notable exceptions) for quick, easy and accurate addition. Here are my seven
basic reference positions:


Reference Positions

Multiply the midpoint of any 5prime by 10 and you have just counted a cluster of ten checkers.


This position shows a 5prime from the 4point to the 8point.
The 6point is the midpoint and the count for these ten checkers = 60 pips (6 × 10.) This is so because 5's and 7's average out to 6's, and 4's and 8's also average out to 6's.
Black = 60.



This is just a 5prime around the 4point plus two checkers on the ace point.
Black = 42.



Five checkers each on the 6 and 8points.
Black = 70.



Two checkers each on the 7 and 8points.
Black = 30.



Five checkers on the 8point.
Black = 40.



Two checkers each on the midpoint and opponent's bar point.
Black = 62.



Two checkers on the midpoint and one on the 14 point.
Black = 40.

These seven reference positions combined with key points and mirrors are the backbone of Cluster Counting.


Key Points

The two key points most often used are the 5point and the 20point (opponent's 5point). The 10, 13 and 15points are also quite valuable.
Using 5Point as Key Point


This position shows two examples of counting a cluster of eight checkers all at once as if they were eight 5's = 40.
Black = 40. White = 40.

Using 20Point as Key Point
The 20point (opponent's 5point) is the most useful key point. All checkers in your opponent's home board should be counted as 20 plus the pips required to get to the 20point.


Black's count is 108 which can be visualized as five 20's + 4 (two each from the 22point to the 20point) + 4 (one from 24point to 20 point).
White's count is 89, visualized as four 20's + 4 + 5 (for the checker on the bar).
Black = 108. White = 89.



Mirrors

Mirrors are another important counting tool. Any point on the board plus its mirroropposite point equals 25. For example, the 5point + 20point, the 1point + 24point, and the 12point + 13point all total 25 pips. It follows that any cluster of 4 checkers in mirror positions total 50.


Black: (20 + 5 = 25) × 2 = 50.
White: (13 + 12 = 25) × 2 = 50.
Black = 50. White = 50.



Black: (23 + 2 = 25) + (24 + 1 = 25) = 50 .
White: (18 + 7 = 25) × 2 = 50.
Black = 50. White = 50.



Mental Shifting

Ok! It would be nice if every time you needed a pip count, the board would consist of clusters as previously described. Unfortunately, that doesn't happen. Fortunately, these easytocount clusters are relatively simple to form by mentally moving the checkers where you want them.
OneWay Mental Shift
Oneway mental shifting involves moving the checkers forward to key points or reference positions and then adding the forward movement to the value of the key points or reference positions.


Black can be easily counted in three clusters: 40 (eight 5's) + 33
(RP4 + 3 pips) + 64 (three 20's + 4) = 137.
Divide White's checkers into three clusters: 44 (5prime + 4 pips forward, 2 each from the 7point to the 5point) + 33 (three 10's + 3 pips from 13 to 10) + 44 (two 20's + 4) = 121.
Black = 137. White = 121.

Note that two of White's checkers were shifted to White's 5point which is occupied by Black's checkers. When shifting one player's checkers, the other player's checker position can be ignored.
TwoWay Mental Shift
Twoway mental shifting differs from oneway mental shifting in that checkers are shifted either forward or backward to key points or reference positions and then compensating shifts are made in the opposite direction on the same side of the board, or in the same direction on opposite sides of the board.


Black's spare checkers on the 6 and 8 points are on the same side of the board. By shifting them one pip in opposite directions to the 7point, a 5prime is formed. Black's position can easily be counted in two clusters: 70 (5prime) + 65 (five 13's) = 135.
White's spare checkers on the 8 and 13points are on opposite sides of the board. By shifting them in the same direction, in this case lefttoright, a 5prime is formed (RP1) and RP7 is also formed. White's position can then be counted in three clusters: 60 + 40 + 42 (two 20's + 2) = 142.
Black = 135. White = 142.

It should be noted that there are often several cluster counting choices available. For instance, in Black's position above, instead of forming a 5prime, you could have shifted the two 9point checkers to the 8point and compensated by shifting the two 5point checkers to the 6point to form RP3. This cluster is also 70 pips.


Your Turn

Let's try counting some positions. Original positions and adjusted positions (after shifting) are shown but not described. Can you spot the shifts? If not, set them up on your backgammon board and they will become clear.
Example 1
Before shifting
 
After twoway mental shifting




Black can be counted in three clusters: 40 (5prime from the 6point to the 2point) + 50 (mirrors on the 7point and the 18point) + 10.
White can be counted in two clusters: 44 (5prime + 4) + 40 (four 10's).
Black = 100. White = 84.
Example 2
Before shifting
 
Black's position after shifting




Black can be counted in three clusters: 30 (six 5's) + 43 (RP5—five 8's + 3) + 84 (four 20's + 4).
Before shifting
 
White's position after shifting




White can be counted in three clusters: 42 (eight 5's + 2) + 40 (RP7) + 67 (three 20's + 7).
Black = 157. White = 149.
Example 3
Before shifting
 
After shifting




Black can be counted in two clusters: 66 (twelve 5's + 6) + 40 (two 20's).
White can be counted in two clusters: 30 (six 5's) + 70 (RP4 × 2 + 10 for two checkers moved from the 13point to the 8point).
Note that in the shifted position White has only 14 checkers. The two checkers originally on the 3point were shifted in different directions—one checker to the 6point and the other checker off the board.
Black = 106. White = 100.
Example 4
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:
With a minimum of shifting, Black's position can be quickly counted in several different ways:
 63 (5prime + 3) + 75 (five 13's + 10 by shifting two checkers from the 18point to the 13point);
 63 (5prime + 3) + 62 (RP6) + 13 (spare checker on the 13point);
 50 (mirrors on the 12 and 13points) + 50 (mirrors on the 7 and 18points) + 30 (six 5's) + 8 (checker on the 8point).
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!


Kevin Bastian created the graphics for this page. Kate McCollough created the original HTML.
See also: Other articles on pip counting.
Return to: Backgammon Galore
