Odds of shutting the box with 9 numbers
The odds of shutting the box with the classic 9 numbers depend on the one-die rule:
| Rule | Best chance of shutting the box |
|---|---|
| One die allowed once 7, 8 and 9 are shut | 9.8% (about 1 in 10) |
| Always two dice | 7.1% (about 1 in 14) |
| One die allowed, but at most two tiles per roll | 7.9% (about 1 in 13) |
The one-die rule helps because the last few small tiles are much easier to match with one die than with two.
Odds of shutting the box with 10 and 12 numbers
The odds of shutting the box fall quickly as boxes get bigger. With two dice throughout and perfect play:
| Tiles | Best chance of shutting the box |
|---|---|
| 1 to 9 | 7.1% (about 1 in 14) |
| 1 to 10 | 4.0% (about 1 in 25) |
| 1 to 12 | 0.36% (about 1 in 276) |
Allowing one die once every tile from 7 up is shut raises these to 5.4% for 10 numbers and 0.47% for 12 numbers. The rules for each are on the 10 numbers and 12 numbers pages.
What are your odds if you do not play perfectly?
Your odds of shutting the box depend a lot on how you choose your tiles. With nine tiles and the one-die rule:
| How you choose | Box shut |
|---|---|
| Perfect play, aiming only to shut the box | 9.8% |
| Fewest tiles, highest first | 9.6% |
| Lowest tiles first | 1.1% (about 1 in 90) |
The simple rule in the middle is easy to remember and loses very little. The Shut the Box strategy guide explains it.
How often each dice total comes up
The odds of each total with two dice explain most Shut the Box strategy. Two dice can land 36 ways, and 7 is the most common total because six of those ways add up to 7:
| Total | Ways out of 36 | Total | Ways out of 36 |
|---|---|---|---|
| 2 | 1 | 8 | 5 |
| 3 | 2 | 9 | 4 |
| 4 | 3 | 10 | 3 |
| 5 | 4 | 11 | 2 |
| 6 | 5 | 12 | 1 |
| 7 | 6 |
Big totals are rare: a 9 or more comes up only 10 times in 36. That is why the 7, 8 and 9 tiles are worth shutting while you can.
How are these odds worked out?
These Shut the Box odds come from going through every possible game. For each set of open tiles, we looked at every roll and every legal choice, and picked the choice that gives the best chance from then on. Doing that for all 512 possible sets of open tiles gives the exact chance from the start. We checked our figures against Durango Bill's published analysis: the 9-number odds with and without the one-die rule, and the 10 and 12-number odds with two dice, agree to the first two decimal places. Our "at most two tiles" figure differs from his because his one-die rule is not the same as ours.
What are the odds in the daily game?
In our daily game, everyone gets the same dice, so the odds change from day to day. Looking ahead with perfect knowledge of the rolls, the dice allow the box to be shut on about 1 day in 4; on the other days nobody can shut it, however well they play. When you finish, you can see the best score our Super Intelligence could get with that day's dice, knowing every roll in advance.