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Dice & Probability

Yahtzee

Expectimax over dice & scorecard states.

1+ players1 solverSolved
ExpectimaxImport stateReplayBest moveProbabilities
Your dice
Tap +/- on each die to set the value. Highlighted dice are the ones the optimal strategy keeps.
Rerolls left
Suggestion
Backed by a precomputed expectimax table — provably optimal under the standard Hasbro joker rule.

Tap Suggest to compute the optimal play.

Deep dive

How BoardSolve plays Yahtzee

Solitaire Yahtzee has a tractable state space (~10^8 reachable states) once you exploit the symmetry of unordered dice multisets. BoardSolve uses a precomputed expectimax table to recommend keep / reroll and category choices that maximise expected score.

State space

A Yahtzee state is the tuple (which of 13 categories are still open, upper-section running total clamped to 63, Yahtzee bonus eligibility, dice multiset, rolls remaining). Dice are unordered, so the multiset has only 252 values per roll instead of 7 776 ordered rolls.

Expectimax alternates max nodes (choose which dice to keep, or which category to score) with chance nodes (the next roll's multiset under the multinomial distribution).

What you get

For every dice roll the solver returns the keep-subset that maximises the expected final score from the current scorecard, and at the end of a turn the category that maximises the same quantity — matching the optimal-play values first published by Tom Verhoeff and James Glenn.

References & further reading

  1. Verhoeff, T. (1999, rev. 2018) . Eindhoven University of TechnologyCanonical computation of the optimal expected score (≈ 254.59) and the expectimax recurrence.
  2. Glenn, J. (2006) . Loyola College Technical Report CS-TR-0002Independent computation matching Verhoeff's values; the basis for cross-checking BoardSolve's table.
  3. Woodward, P. (2003) . Chance 16(1), 18–22