About
These rules give accurate cube decisions for 3 vs. 3 checker bearoff positions. As with 4 vs. 4 checker bearoffs, generic race formulas like the Keith count and even more advanced methods like EPC don't work well in such short races, so specialized tools are needed.
These rules are based on Art Benjamin's "Math Overboard: Three Checkers Left and Variations" (PrimeTime Backgammon, Winter 2021). The article discusses 3 vs. 3 checker positions in much greater detail; this page contains the minimum information needed to apply the rules over the board.
When to use
Use these rules to calculate cube decisions when both sides have exactly 3 checkers left to bear off.
How to use
Step 1: Calculate each side's adjusted pip count (APC).
Start with each side's pip count, then add penalties for spare checkers:
APC = pip count + penalties:
| Condition | Penalty | Interpretation |
|---|
| 1-point spares | +2 each | 1-point spares are very bad |
| 2-point spares | +1 each | 2-point spares are somewhat bad |
| 3–6 point spares | +0.5 each | Higher-point spares are slightly bad |
Step 2: Compare APCs for leader vs. trailer to get the cube decision.
Let X be the leader's (doubler's) APC and Y be the trailer's (taker's) APC:
| Leader's APC (X) | Rule | Interpretation |
|---|
| X < 11 | Always double/pass | Too far ahead to take |
| X = 11–11.5 | Always double, take if Y < X | Take only if strictly ahead on APC |
| X = 12–12.5 | Always double, take if Y ≤ X | Take if equal or ahead on APC |
| X = 13–14.5 | D/T window: X-3 ≤ Y ≤ X+1 | Double within 3 pips, take up to 1 pip worse |
| X ≥ 15 | D/T window: X-2≤ Y ≤ X+1 | Double within 2 pips, take up to 1 pip worse/td> |
Tips
1. APC penalties vs. Keith count penalties
The 1- and 2-point penalties are the same as Keith count. The difference is that 3–6 point spares are +0.5 each (Keith count has different rules for the 3-point and empty 4/5/6 points).