There are good similarities among three different pip counting systems, Douglas Zare's "Half-Crossover Pipcount", Nack Ballard's "Naccel", and my system. Those three systems separate all 26 points of the board in a number of groups, assign a group center, and provide error numbers from the center point to locate any points in a group.My system, the "Five-Count," is described here.
A detailed explanation for Douglas Zare's "Half-Crossover Pipcount" is available at this link
Nack Ballard's "Naccel" is explained in a GammOnLine article. Note that the "Naccel" shown in GammOnLine is slightly different from the one explained in Backgammon Today Sep. 2001 issue, but is the "bumped" one introduced in the magazine. The "Naccel" I referred in this page is the one with not bumped quads (the 6 point has 6 extra pips.)
To show "summation" operation, I use a "sigma" sign, Σ, as follows:
_{15} | ||||
Σa_{i} | = | Σa_{i} | = | a_{1} + a_{2} + a_{3} + ... + a_{14} + a_{15} |
^{i=1} |
I also use these symbols:
x_{i} | = | A pip count of a checker located any point. i = 1, 2, ... , 15 |
P | = | A total pip count. |
x_{i} = 5g_{i} + e_{i}
P = Σ(x_{i}) = 5Σg_{i} + Σe_{i}
where
g_{i} = 0, 1, 2, 3, 4, 5 (group)
e_{i} = −2, −1, 0, +1, +2
x_{i} = 5 + 3h_{i} + e_{i}
P = Σ(x_{i}) = 75 + 3Σh_{i} + Σe_{i}
where
h_{i} = −2, −1, 0, 1, 2, 3, 4, 5, 6, 7 (half crossover)
e_{i} = −1, 0, +1
Note: the error number of the goal point (or the pocket) is +1 of h = −2.
x_{i} = 6q_{i} + e_{i}
P = Σ(x_{i}) = 6Σq_{i} + Σe_{i}
where
q_{i} = 0, 1, 2, 3, 4 (quad)
e_{i} = +1, +2, +3, +4, +5, +6Since all e_{i} is positive (+) and Σe_{i} often becomes too big to handle for a human player, another breakdown in the Σe_{i} part is called for:
Σe_{i} = 6s + r
P = 6(Σq_{i} + s) + r
where
s = 0, 1, 2, ..., 15 (squad)
r = 0, 1, 2, ... (remains, usually r < 6)
x_{i} = 6q_{i} + e_{i}
P = Σ(x_{i}) = 6Σq_{i} + Σe_{i}
where
q_{i} = 0, 1, 2, 3, 4 (quad)
e_{i} = 0, +1, +2, +3, +4, +5
x_{i} = 6 + 6q_{i} + e_{i}
P = Σ(x_{i} ) = 90 + 6Σq_{i} + Σe_{i}
where
q_{i} = −1, 0, 1, 2, 3
e_{i} = 0, +1, +2, +3, +4, +5
x_{i} = 6 + 6q_{i} + e_{i}
P = Σ(x_{i} ) = 90 + 6Σq_{i} + Σe_{i}
where
q_{i} = 0, 1, 2, 3
e_{i} = 0, +1, +2, +3, +4, +5 (for q_{i} = 1, 2, 3)
e_{i} = −6, −5, ... , −1, 0, +1, ... , +5 (for q_{i} = 0)Note: the error number of the goal point (or the pocket) is −6.
x_{i} = F + Bg_{i} + e_{i}P = Σ(x_{i} ) = 15F + BΣg_{i} + Σe_{i}
Parameters B
(base)F
(offset)g_{i} (group) e_{i} (error) Five-Count 5 0 0, 1, 2, 3, 4, 5 −2, −1, 0, 1, 2 Half-Crossover 3 5 −2, −1, 0, 1, ... , 7 −1, 0, 1 Naccel 6 0 0, 1, 2, 3, 4 1, 2, 3, 4, 5, 6 Bumped Naccel 6 0 0, 1, 2, 3, 4 0, 1, 2, 3, 4, 5 HN Naccel 6 6 −1, 0, 1, 2, 3 0, 1, 2, 3, 4, 5, HF Naccel 6 6 0
1, 2, 3−6, −5, −4, ... , 4, 5
0, 1, 2, 3, 4, 5
See: Other articles by Sho Sengoku
See: Other articles on Pip Counting
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