Stack and Straggler
by Douglas Zare
2 February 2015
Backgammon games are often concluded by contact bearoffs. We often have a checker sent back in the bearoff, and need to evaluate a race with one checker coming home. If the checkers in your home board are on the ace point, this is a classic stack-and-straggler position. There is a formula due to Walter Trice for estimating the effective pip counts of stack-and-straggler positions: 3.5 pips times the number of checkers (including the straggler) plus the position of the straggler. In this column, we will review Trice's formula, and we will consider more complicated home board positions with one straggler. (In "Stack and Stragglers", we looked at having more than one straggler.)
For example, suppose Red has 5 checkers on the ace point and a straggler on the 17 point (see this discussion).
White redoubles. |
What is Red's epc?
Red has a total of 6 checkers, so Trice's stack-and-straggler formula estimates the epc as 6 * 3.5 + 17 = 21 + 17 = 38. The actual epc is 38.078, so Trice's formula is accurate to within a tenth of a pip here.
Walter Trice also observed that the effective pip count of a pure n-roll position is close to 7n+1, and this formula gets increasingly accurate as n increases. For n=7, Trice's 7n+1 formula is low by only 4 millionths of a pip. It was a surprise that a simple formula was so accurate, and we can't expect the same accuracy for a simple stack-and-straggler formula. We can expect is that there is some constant x_5 so that with 5 checkers on the ace point, and a straggler on point p, the effective pip count is about x_5+p, and for the right choice of constant x_5, this formula gets increasingly accurate. It can be off by a significant amount when p is small, say p=7, since it is sometimes quite inefficient to bear in from the bar point. We can't expect x_5 to be a whole number. Trice's stack-and-straggler formula suggests using x_5 ~ 6 * 3.5 = 21, and the actual value is about 21.042.
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