About
These rules give accurate cube decisions for 4 vs. 4 checker bearoff positions. As with 3 vs. 3 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.
This approach is based on Art Benjamin's article "Math Overboard: Four Checkers Left" (PrimeTime Backgammon, Winter 2022). The article discusses 4 vs. 4 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 4 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 and position adjustments:
APC = pip count + checker penalties + adjustments:
Checker penalties:
| Condition | Penalty | Interpretation |
|---|
| First 1-point spare | +2 | First spare on ace point |
| Additional 1-point spares | +1 | Additional ace-point spares less bad |
| 2-point spares | +1 each | |
| 3–6 point spares | +0.5 each | |
Adjustments:
| Condition | Adjustment | Interpretation |
|---|
| Both 4-pt and 5-pt empty | +0.5 | Gap on high points |
| Both 4-pt and 5-pt occupied | −0.3 | Good distribution bonus |
| 1-pt occupied and pips ≥ 13 | +0.3 | Ace-point penalty for longer positions |
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 | Race too short for trailer to catch up |
| 11 < X < 13 | D/T window: 11 ≤ Y ≤ X+1 | Double if Y ≥ 11, take up to 1 worse |
| 13 ≤ X < 17.7 | D/T window: X−2 ≤ Y ≤ X+1 | Double within 2, take up to 1 worse |
| X ≥ 17.7 | D/T window: X−2 ≤ Y ≤ X+1.5 | Double within 2, take up to 1.5 worse |
Tips
1. Differences from 3v3 APC
The 2-point and 3–6 point spare penalties are the same as 3v3. The differences are: (1) the 1-point penalty is +2/+1/+1 for 2nd/3rd/4th checkers instead of a flat +2 per spare, and (2) there are three position-based adjustments (gap, both-occupied, ace-point) that don't exist in 3v3.