Shut the Box Strategy and Odds

How these odds were worked out

A 1–9 box has only 512 possible sets of open tiles, so a computer can work out the best play exactly. Starting from a shut box and working back, it looks at every position, every roll and every set of tiles that roll could shut, and keeps the choice that gives the highest chance of shutting the box. It does the same for the lowest expected score. The game above runs the same calculation in your browser, which is what its Hint button and odds line use. All the rules here are the standard ones: shut any combination that adds up, and one die only once the 7, 8 and 9 are down.

The odds of shutting the box

So even perfect play shuts a classic box about once in ten games. The average score is for play aimed at the lowest score, which is not quite the same as play aimed at shutting the box: when shutting it is no longer possible, the best move is the one that leaves the least behind.

The best first move for every roll

On a full 1–9 box, these are the tiles to shut for each roll, and your chance of shutting the box afterwards with best play. The last column is the next best choice, to show how much the decision matters.

The pattern is clear: shut the single tile when you can, and from 10 up, shut the 9 with the smallest tile that makes up the difference.

Two rules that get you nearly all the way

1. Shut as few tiles as you can, and the highest ones. A 9 can only be shut by a roll of 9 or more, while a 1 or a 2 can be part of almost any roll, so keep the small tiles for later. Of the ways to make the roll, pick the one with the fewest tiles, and if there is a choice, the one with the highest tile.
2. Once 7, 8 and 9 are down, roll one die only if the open tiles add up to 6 or less. This rule is exact: in every one of those positions on a 1–9 box, one die is the better choice when the open tiles add up to 6 or less, and two dice are better otherwise.

Here is how often each way of playing shuts a 1–9 box, against the 9.76% of perfect play:

The two simple rules come close to perfect play. Shutting as many tiles as possible, which feels like progress, is the worst habit of all: it spends the small tiles early and leaves high ones that only a big roll can shut.

Why the one-die rule matters

Two dice can never make less than 2, and most often make 6, 7 or 8. When only small tiles are left, that is often too much, and one die, which makes 1 to 6 with equal chance, fits better. When the open tiles still add up to more than 6, one die cannot finish the job in a single roll and two dice give more ways to continue.

Playing for the lowest score

When you are playing against others for the lowest score, the aim changes once shutting the box is out of reach: get rid of as much as possible. The Hint button handles both: it picks the move with the best chance of shutting the box, and when no move can do that, the one with the lowest expected score.