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Counting Pips Quickly Using Bob Koca's Shortcut

Posted By: Paul Weaver
Date: Tuesday, 7 September 2010, at 3:24 a.m.

Three diagrams are shown in this post.

Not content to rest on my laurels after winning in Baden, I have been busy exploring theoretical topics in backgammon. I will share some of my research with bgonline readers here.

Believe it or not, while playing a few days ago, I noticed that I reached the same position (shown below) in a non-contact race thirty games in a row. Opponent's position was different (by coincidence) in every game and is not shown.

After counting the position thirty times, I realized that I could save a lot of time by applying Dr. Bob Koca’s algebraic shortcut. (a + b) (a - b) = a x a – b x b.

My extensive analysis continues below the diagram.

is Player 2

score: 0
pip: 0
Money session
Jacoby Beaver
pip: 85
score: 0

is Player 1
XGID=---ABCDE------------------:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

Here is how I computed the pip count. (Bob, please let me know if you agree with my application of your formula.)

To get the pip count of the 5 checkers on the 7pt, (6 + 1) (6 – 1) = (6 x 6) – (1 x 1) =36 -1 = 35.

To get the pip count of the 4 checkers on the 6pt, (5 + 1) (5 – 1) = (5 x 5) – (1 x 1) =25 -1 = 24.

To get the pip count of the 3 checkers on the 5pt, (4 + 1) (4 – 1) = (4 x 4) – (1 x 1) = 16 -1 = 15.

To get the pip count of the 2 checkers on the 4pt, (3 + 1) (3 – 1) = (3 x 3) – (1 x 1) =9 -1 = 8.

Last but not least, to get the pip count of the 1 checker on the 3pt, (3 + 1) (3 – 1) = (2 x 2) – (1 x 1) =4 -1 = 3.

Now, just add 35 + 24 + 15 + 8 + 3 = 85.

Thanks, Bob, for the terrific shortcut!

In the next thirty games of the session, believe it or not, the following position came up (discussion continues after the diagram).

is Player 2

score: 0
pip: 255
Money session
Jacoby Beaver
pip: 195
score: 0

is Player 1
XGID=--------o----O------------:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 94.07% (G: 17.79% B: 0.45%)
Opponent Winning Chances: 5.93% (G: 0.12% B: 0.00%)
Cubeless Equities
No Double:+0.881
Double:+1.655
Cubeful Equities
No Double:+1.000 (0.000)
Double/Take:+1.554 (+0.554)
Double/Drop:+1.000
Best Cube action: Double / Drop

eXtreme Gammon Version: 1.21

With 15 checkers on my 13pt, my pip count was 14 x 14 – 1 x 1 = 196 -1 = 195. With 15 checkers on his 17pt, my opponent’s count was 16 x 16 – 1 x 1 = 256 – 1 = 255. Ahead by 60 pips, I decided to double and my opponent took. I then rolled 55 twenty times in a row and could not move. Then I rolled 65 twelve times in a row and was forced to play 13/2 each time, reaching the following position (discussion continues after diagram).

is Player 2

score: 0
pip: 89
Money session
Jacoby Beaver
pip: 63
score: 0

is Player 1
XGID=--L-----b---aC-----bbbbbb-:1:-1:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

I then rolled 32, playing 13/10 13/11. Opponent rolled 11, hit three checkers and won a backgammon. Was I unlucky? I should tell you the exact same sequence occurred thirty games in a row.

What are the chances of this occurring thirty games in row? Very very slim, perhaps 1/30. Showing off my match skills, I will mention that 1/30 = 3.3333333%.

Congratulations to Bob for discovering a very useful formula and thanks to Bob for sharing his formula with us!

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