Shut the Box strategy and odds

The solved numbers

These are the exact results of playing perfectly. There are two different perfect strategies because there are two different goals — minimising your score, and maximising your chance of shutting every tile — and they do not give the same answer.

The gap between the two objectives is small but real: chasing a perfect game costs about 0.44 points of average score on the nine-tile board and buys roughly half a percentage point of extra perfect-game chance. If you are playing a scored game, minimise your score. If you are playing pub rules where only shutting the box matters, take the other line.

The two strategies disagree more often than you would think

On the nine-tile board the two optimal strategies choose different moves in 267 distinct positions; on the twelve-tile board they differ in 1,098. The clearest example happens on the very first roll.

Both lines shut two tiles worth ten points, so they look equivalent — but they are not. Taking 1+9 clears the awkward 9 and banks the safest possible single point, which is what you want when every point counts. Taking 4+6 instead keeps the small tiles 1, 2 and 3 available, and those small tiles are what let you finish the last few points of a perfect game. The nine is a liability for your score and an asset for your perfect-game chances.

For every roll other than 10 and 11 on a full nine-tile board, both strategies simply shut the single tile matching your roll.

Practical rules of thumb

You will not memorise 512 positions at the pub. These heuristics capture most of the value of optimal play, and you can check any of them against the hint engine built into the game on this site.

1. Shut the high tiles early. A 9 left open late is nine points and needs a specific roll to clear; a 1 left open is one point and clears easily.
2. Get under the single-die threshold. Once every open tile is 6 or lower you may roll one die, which sharply improves your chance of matching. Moves that clear 7, 8 and 9 quickly are worth more than they look.
3. Prefer fewer tiles when the points are equal — with one exception. Shutting two tiles instead of three usually leaves you more combinations, but keep the small tiles 1, 2 and 3 if you are chasing a perfect game, because they are what finish it.
4. Avoid leaving a lone 7, 8 or 9. With two dice a single high tile can only be matched by that exact total, and 7 is the only total that comes up more often than one roll in seven.
5. On the 12-tile board, accept that a perfect game is rare. At roughly 1 in 268 even with perfect play, the twelve-tile board is a scoring game, not a shut-the-box game.

How these numbers were computed

Every figure on this page is reproducible. The game is solved by backward induction: the value of a position is the probability-weighted average over all dice rolls of the best move available from it, evaluated recursively down to the empty board.

• Every subset of open tiles is a state — 512 states on the nine-tile board, 4,096 on the twelve-tile board.
• Two-dice totals are weighted by their real probabilities (7 comes up 6 times in 36; 2 and 12 once each), and single-die rolls are weighted 1 in 6 each.
• A roll with no legal combination ends the turn, and that state contributes its own tile sum to the average.
• The same solver runs in your browser on this site, so the hint engine gives the genuinely optimal move for your exact position rather than a rule of thumb.
• Assumptions: two six-sided dice, one combination shut per roll, the single-die rule once all open tiles are 6 or lower, and scoring as the sum of open tiles. Different house rules change the numbers.

Strategy questions

What are the odds of shutting the box?

With optimal play on the nine-tile board, about 7.4% — roughly one game in 13 — if you play specifically to maximise that chance, or 6.95% (one in 14) if you play to minimise your score instead. On the twelve-tile board it is about 0.39%, or one game in 254. Casual players who shut tiles without thinking do considerably worse than these figures.

How do you win at Shut the Box every time?

You cannot. Shut the Box is a dice game, and even flawless play shuts the box only about once in 13 or 14 games on the nine-tile board. What optimal play does guarantee is the best possible average — 11.66 points on nine tiles. Any claim of a method that always wins is false.

Is it better to shut one big tile or several small ones?

Usually the bigger tile, because high tiles are hard to clear later and cost the most if left open. The exception is when you are playing for a perfect game: keeping 1, 2 and 3 available is what lets you finish the last few points, which is why optimal perfect-game play shuts 4+6 rather than 1+9 on an opening roll of 10.

Does the single-die rule really matter?

Substantially. Rolling one die gives you six equally likely totals from 1 to 6, all matchable by a single small tile, instead of eleven totals from 2 to 12. Reaching that position with tiles still open is one of the main goals of good play, which is why clearing 7, 8 and 9 early is worth more than their face value suggests.