Roll one die and keep adding. Roll a 1 and the whole turn is gone. Stop whenever you like — and knowing when is the entire game.
- 53.06 % — how often the first player wins, both sides perfect
- 21 the right stopping point at level scores, not 20
- 49 the right stopping point eighty points behind
- 4 points is all that perfect play is worth over the folk rule
How do you play Pig?
- Roll one die and add it to your turn total. Keep rolling as long as you like.
- Roll a 1 and you lose everything from this turn, and the die passes over.
- Stop whenever you want and your turn total goes on your score.
- First to 100 wins.
That is the complete rulebook. One die, no board, and a decision on every single roll: the pot grows, and so does the chance of losing it.
When should you stop rolling?
A position is your score, their score, and the points riding on this turn — about a million of them, which is small enough to evaluate every one exactly rather than argue about it.
At level scores the answer is 21. So the folk rule is close enough that nobody should feel foolish for using it.
The game will simply tell you, if you want it to. Switch on Show the solved answer and every turn comes with the right call for that exact score.
What is thinking worth at Pig?
Four percentage points. Playing the exact solution against someone who always stops at 20 wins 54 games in 100. Against a nervous player who stops at 15 it is 59.
That is the honest ceiling, and it is worth stating plainly: Pig is a game you can play perfectly and still lose, often. The die has the rest — which is exactly why it survives as a family game while stronger games of skill do not.
Who has the advantage — first or second?
The first player, 53.06 %, with both sides playing perfectly. It is not nothing: over a long evening it is worth about one game in twenty.
That number is also how we know the arithmetic here is right. It matches the published solution of Pig exactly, and a computation that lands on somebody else's answer to two decimal places is a computation you can trust.
How strong is the computer?
| Level | How it plays | Against the next one down |
|---|---|---|
| Easy | stops at 15 | — |
| Medium | stops at 20, the table rule | 55.6 % |
| Hard | the exact stopping point for every score | 54.2 % |
The hard level is not better because it stops one point later. It is better because it stops at a different point depending on the score — pushing when behind, banking early when ahead. A fixed number cannot do that, and that single difference is the whole four points.
Where does "stop at 20" come from?
It is not folklore. It is the exactly correct answer — to a different question.
Ask "which fixed stopping point banks the most points per turn?" and the answer is 20, worth 8.14 points a turn. Stop at 15 and you average 7.85; stop at 25 and you average 8.00. Twenty is the peak, and it is a real peak: the rule is arithmetic, not superstition.
| Stop at | 15 | 18 | 20 | 22 | 25 | 30 |
|---|---|---|---|---|---|---|
| Points a turn | 7.85 | 8.09 | 8.14 | 8.13 | 8.00 | 7.58 |
But points per turn is not what you are trying to win. You are trying to reach 100 before somebody else does, and those two goals only agree while the scores are level and the finish is far away. As soon as one player is close to home, collecting points efficiently stops being the point — the question becomes whether you can get there first, and the answer to that changes every turn.
That is the whole gap between the folk rule and the solved one, and it is why the map above is not flat.
Where does Pig come from?
Nobody knows, which is normal for a folk dice game: it needs one die and no equipment, so it has almost certainly been played in some form for as long as dice have existed. It appears in print in twentieth-century collections of family games, usually with the same rules and occasionally under a different name.
What is documented is its second life. Pig became the standard worked example for teaching expected value and dynamic programming, and the complete solution was published in that context — which is why a children's dice game has a peer-reviewed optimal strategy at all.
Why is Pig in every programming course?
Because the rules fit in four lines and the interesting question does not.
A first-year student can write the game in an afternoon. Working out when to stop needs expected value, then a recursion over a million states that refers to itself, then the realisation that the answer depends on the opponent's score as well as your own. It is the shortest path from a toy to a real problem, and that is why it turns up in lectures far more often than it turns up on kitchen tables.
More dice here
Farkle asks the same question with six dice and a scoring table, Shut the Box asks a different one, and dice games for two players compares seven of them by how much skill is actually in each. If you want to keep score for a table with real dice, the dice score sheet is here too.
