Backgammon Doubling Cube Calculator
Work out backgammon doubling-cube decisions, with the 25 per cent take point derived rather than quoted — and why the live cube moves it.
The 25% take point, derived
Pass a double and you lose 1 point. Take it and the cube sits at 2, so you win 2 points with probability p and lose 2 otherwise — equity 4p − 2. Taking is at least as good as passing when 4p − 2 ≥ −1, which is p ≥ ¼. Sweeping p numerically rather than trusting the algebra puts the crossing at 25.0%, exactly where it should be.
The same algebra gives the other half. Holding the cube at 1 is worth 2p − 1; doubling into a take is worth 4p − 2; the second wins when p ≥ ½ — the sweep agrees at 50.0%. So a dead cube says: double the moment you are ahead, and take down to a quarter.
But nobody plays with a dead cube
Take a double and you own the cube — you may redouble later, and that option has value. Put it in and both numbers move, one of them enormously.
| Model | Take point | Doubling point |
|---|---|---|
| Dead cube | 25.0% | 50.0% |
| Live cube | 20.0% | 80.0% |
The take point drifts 5 points. The doubling point moves 30. "Double as soon as you are ahead" is not a small simplification of correct play — it is a different strategy, and the continuous model says hold until four times out of five.
Both figures are derived, not quoted. Between cube decisions equity is a martingale, so it is linear in p: owning the cube it runs straight from (0, −1) to (D, +1), giving E(p) = −1 + 2p/D. Indifference to a take is 2E(p) = −1, so p = D/4; and you double exactly when your opponent is at their take point, so D = 1 − TP. Hence TP = (1 − TP)/4 = ⅕.
| Grid size | Sweeps to converge | Take point | Doubling point |
|---|---|---|---|
| 100 | 13,318 | 21.0% | 80.0% |
| 200 | 49,016 | 20.5% | 80.0% |
| 400 | 178,838 | 20.3% | 80.0% |
An independent value iteration, which knows nothing of the algebra, walks towards the same answer as the grid is refined — and lands on the doubling point exactly at every size. Worth noting how slowly it converges: a first attempt stopped early and reported a take point of 24.7%, which is simply the starting guess barely moved.
And gammons do not cancel
A win is worth 1 + gw points on average and a loss 1 + gl, so the take point becomes (1 + 2gl) / (4 + 2gw + 2gl) — which collapses back to a quarter when neither side wins gammons.
| Gammon share of wins | Gammon share of losses | Take point |
|---|---|---|
| 0% | 0% | 25.0% |
| 0% | 10% | 28.6% |
| 0% | 25% | 33.3% |
| 15% | 0% | 23.3% |
| 15% | 15% | 28.3% |
| 30% | 10% | 25.0% |
Gammon losses push the take point up — there is more to lose by taking — and gammon wins pull it down.
But they do not cancel when they are equal, which is the tempting shortcut. At 15% each way the take point is 28.3%, not 25%. Passing costs exactly 1 point however gammonish the position is, while taking exposes you to 2(1 + gl) — gammons inflate one side of the comparison only, so they make taking scarier even when both players are equally likely to win one.
What actually holds the quarter is gammon wins running at three times gammon losses. Put gw = 3gl into the formula and it becomes (1 + 2gl) / (4 + 8gl) = (1 + 2gl) / 4(1 + 2gl) = ¼, for every value of gl. The factor cancels exactly.
Which is why "the take point is 25%" needs its conditions attached. It is exactly right for a cube that can never be turned again, in a game with no gammons. Change either and it moves.
How to use
- Enter your winning chance as a percentage.
- Add the share of your wins and losses that are gammons, if you know them.
- Pick a live cube if redoubles are allowed, which is the normal game.
- Read off whether to take a double and whether to offer one.
- Compare the two models to see how much the redouble option is worth.
Frequently asked questions
Where does the 25 per cent take point come from?
From two lines of algebra. Passing costs you exactly 1 point. Taking puts the cube at 2, so you win 2 with probability p and lose 2 otherwise, giving equity 4p minus 2. Taking is at least as good as passing when 4p minus 2 is at least minus 1, which is p at least a quarter.
Should I really double as soon as I am ahead?
Only with a dead cube. The same algebra says doubling beats holding when p is at least a half, but that assumes nobody can ever redouble. Once the taker owns the cube and may turn it later, the doubling point rises to about 80 per cent — a thirty point difference, which is a different strategy rather than a refinement.
What is the live-cube take point?
A fifth, or 20 per cent. Between cube decisions equity is a martingale and so runs in a straight line, which makes the model solvable by hand: owning the cube your equity goes from minus 1 at zero to plus 1 at the doubling point D, you are indifferent to a take when equity is minus a half, giving a take point of D over 4, and you double exactly when your opponent reaches their take point so D is 1 minus the take point. Solving gives a fifth.
How do gammons change the take point?
The take point becomes (1 + 2gl) divided by (4 + 2gw + 2gl), where gw and gl are the shares of your wins and losses that are gammons. Gammon losses raise it because there is more to lose by taking; gammon wins lower it.
Do gammons cancel out if both sides win them equally often?
No, and this catches people. At 15 per cent each way the take point is 28.3 per cent, not 25. Passing costs exactly 1 point however gammonish the position is, while taking exposes you to twice (1 + gl), so gammons inflate only one side of the comparison. What actually holds the quarter is gammon wins running at three times gammon losses — substitute that in and the factor cancels exactly.
Does this send anything anywhere?
No. Everything is computed in your browser and nothing is uploaded.
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