Dominós Strategy: Triangular Numbers

A successful and comprehensive dominoes strategy utilizes functions involving triangular numbers.

example: T4=10
.
..
...
....

T1=1, T2=3, T3=6, T4=10

Tn= [n(n-1)]/2

Applied to dominoes:

Quantity of tiles in a double set; where n = highest pip number
[(n+1)(n+2)]/2
example: for a double-six set, n=6 yielding 28 tiles

Total pip quantity in a double set; where n = highest pip number
[n(n+1)(n+2)]/2
example: for a double-six set, n=6 yielding a total of 168 pip (168 “dots” in a double-six set)

Highest possible pip quantity from a 7-piece draw; where n = highest pip number
1(n+n)+1(n+(n-1))+2(n+9n-2))+3(n+(n-3)) = (14n-14)
example 1: for a double-six set, n=6; thus, the highest possible pip count with 7 pieces is 70
example 2: for a double-nine set, n=9; thus, the highest possible pip count with 7 pieces is 112

No matter the set, 15 will always be the lowest possible pip count in a 7-piece draw.

With these numbers we can apply the high and low pip count to a normal distribution scale to find the probability of drawing a certain pip count in a 7-piece draw.

standard deviation for a 7-piece draw = 16.30951
mean for a 7-piece draw = 42.5

On a normal distribution scale:
100% probability that someone will draw a pip count between 15-70.
34% probability that someone will draw a pip count between 26-42.
34% probability that someone will draw a pip count between 42-59.
13% probability that someone will draw a pip count between 15-26.
13% probability that someone will draw a pip count between 59-70.

Applying these numbers to a strategy for a Block Game: Clearly, if you draw a pip count of 15, your opponent must have a higher pip count; therefore, you would attempt to block the game as soon as possible. If you draw a pip count of 70, you would alter your strategy to extend the line of play for as long as possible in order to avoid an early block. Basically, if you draw a pip count between 15-26, you want to play for an early block. Also, if you draw a pip count between 59-70, you would want to extend the line of play for as long as possible. Theoretically, you would also have a probable pip advantage if you draw between 26-42 and a probable disadvantage between 42-59.