The maths of Shut the Box
Shut the Box is small enough to solve exactly. Every set of tiles that can be up is a position — 512 of them in the nine-tile box, 4,096 in the twelve — and, working back from a shut box, each is worth the average, over every way two dice can fall, of the best position that roll’s covers lead to. No simulation and no rules of thumb: these are the figures Clack’s engine works out, the same ones the app shows you as you play.
With perfect play
| Box | Chance to shut | Average score |
|---|---|---|
| 9 tiles single-die rule off | 7.14% 1 in 14 | 11.16 |
| 9 tiles single-die rule on | 9.76% 1 in 10 | 11.02 |
| 12 tiles single-die rule off | 0.36% 1 in 276 | 35.08 |
| 12 tiles single-die rule on | 0.47% 1 in 211 | 35.08 |
Each column assumes play aimed at it. The chance to shut comes from playing to shut the box; the average score — the tiles left up at the end, added together — from playing for the lowest score. Usually that is the same play, but not always: see shutting or scoring.
The single-die rule
Once the tiles still up add up to 6 or less, the traditional rule lets you roll one die instead of two. It is a choice, not an obligation — and the solver shows it is never the wrong one. From every position where it is offered, in either box, one die is at least as good as two, for shutting the box and for score.
It is worth 2.62 percentage points to a fresh nine-tile box — 7.14% without it, 9.76% with it — and 0.11 to the twelve-tile box, from 0.36% to 0.47%.
“Take the biggest tiles”
The rule of thumb most people play by: cover with the biggest tiles you can — a lone 9 rather than 6 + 3, 7 + 1 rather than 6 + 2 — and roll one die whenever the rule allows. It sounds like sense, because big tiles are the hard ones to cover later, and most of the time it is right. Here is how often it is not, counting every position and every roll with more than one way to cover it.
| Box | Wrong at | Chance to shut | Average score |
|---|---|---|---|
| 9 tiles single-die rule off | 167 of 1,774 | 6.96% perfect play 7.14% | 11.37 perfect play 11.16 |
| 9 tiles single-die rule on | 190 of 1,774 | 9.48% perfect play 9.76% | 11.24 perfect play 11.02 |
| 12 tiles single-die rule off | 1,296 of 16,747 | 0.35% perfect play 0.36% | 35.35 perfect play 35.08 |
| 12 tiles single-die rule on | 1,406 of 16,747 | 0.45% perfect play 0.47% | 35.34 perfect play 35.08 |
Over a whole game from a fresh nine-tile box, the rule of thumb costs 0.28 percentage points of shut chance with the single-die rule and 0.18 without it. Small, because most of its calls are right — but the ones it gets wrong can be expensive.
Its worst mistake
With the single-die rule, 1 2 3 5 6 7 up and a roll of 11: the biggest tiles are 7 + 3 + 1, which leaves 10.78% to shut the box. 6 + 5 leaves 21.12% — 10.34 percentage points given away on one roll.
Biggest tiles: 7 + 3 + 1
leaves 2 5 6 · 10.78% to shut
Best: 6 + 5
leaves 1 2 3 7 · 21.12% to shut
Without the rule, the worst is 1 2 4 5 6 8 up and an 11: 8 + 2 + 1 leaves 6.64%, where 6 + 5 leaves 11.60%. The twelve-tile box’s worst mistakes are these same two positions: what a position is worth depends only on the tiles up and the rule, not on which box it came from.
The biggest swing
Of every decision in the classic box — nine tiles, single-die rule — this is the one with the widest gap between the best cover and the worst: 1 2 3 6 9 up, a roll of 9. Covering the 9 leaves 24.15% to shut the box; 6 + 2 + 1 leaves 5.25%. That is a swing of 18.90 percentage points on a single roll, with 6 + 3 at 8.46% in between.
It is not a mistake the rule of thumb makes — the 9 alone is the biggest cover, and the right one. It is simply the roll where the choice is worth the most.
Best: 9
leaves 1 2 3 6 · 24.15% to shut
Also legal: 6 + 3
leaves 1 2 9 · 8.46% to shut
Worst: 6 + 2 + 1
leaves 3 9 · 5.25% to shut
| Box | Up, and the roll | Best | Worst |
|---|---|---|---|
| 9 tiles single-die rule off | 1 2 4 5 6 9 rolled 9 | 9 12.57% | 6 + 2 + 1 1.85% |
| 9 tiles single-die rule on | 1 2 3 6 9 rolled 9 | 9 24.15% | 6 + 2 + 1 5.25% |
| 12 tiles single-die rule off | 1 2 3 4 5 12 rolled 12 | 12 14.37% | 5 + 4 + 2 + 1 0.31% |
| 12 tiles single-die rule on | 1 2 3 4 5 12 rolled 12 | 12 24.48% | 5 + 4 + 2 + 1 0.62% |
Shutting or scoring
Shut the Box is scored two ways, and they are not quite the same game. Played all or nothing, only a shut box counts. Played for points, every tile left up counts against you, so getting most of the way is worth something. Usually the same cover is best for both.
Not always. In the classic box, at 52 of 1,774 decision points no single cover is best for both, and 85 with the rule off; with twelve tiles, 587 of 16,747 with the rule and 599 without. The clearest such split in the classic box: 2 3 4 5 6 7 8 up and a 9. 5 + 4 gives the best chance to shut the box, 6.49%, but averages 13.10; 7 + 2 shuts it 6.08% of the time and averages 12.67.
Coach ranks covers by the chance to shut the box, which is how Shut the Box is usually won. When a different cover would have scored lower, the review after the round says so — as a note about score, never as a mistake.
What should I cover?
The calculator runs a port of Clack’s solver in this page. Every time the site is built, the port is checked against the app’s own engine — every position, in both boxes, with the rule on and off — so it gives the figures the app gives.