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Yahtzee Best Move Advisor, Explained

After reading this you will know how the advisor finds the mathematically best dice to keep, how expectimax works on the 252 distinct dice outcomes, and how to read the ranked results without being fooled by near-ties.

What the advisor does

Yahtzee looks like a game of luck, and it is. But at any single decision point the luck is fully bounded. You have five dice showing, one or two rerolls left, and a fixed set of open categories. There are only so many ways the future can unfold, and a computer can count all of them.

The advisor does exactly that. It takes your five dice, the number of rerolls remaining, and the categories still open on your card. Then it computes the expected value (EV) of every legal keep decision by looking at every reroll that could follow, and it tells you which keep gives the most points on average.

Here is the hook. Suppose you roll 1-2-3-4-6 on your first throw with everything open. Your eye says "keep the run, chase the large straight." The advisor agrees, but it also shows you the exact number: keeping 1-2-3-4 and rerolling the 6 is worth about 13.4 points on average, while keeping all five and hoping the 6 becomes a 5 next time is worth less because you have burned a reroll on a single die. The ranking makes the trade-off concrete.

When to use it, and when not

Use the advisor to study. Set up positions that puzzled you in a real game and see what the exact math prefers. It is a training tool, not something you would consult mid-turn at the table.

Do not use it as a full-game solver. The advisor optimizes the current hand in isolation. It maximizes points scored into the best open category right now. It does not plan across turns, and it does not track the 35-point upper-section bonus as part of its number. That single-turn scope is a deliberate simplification, and it is the source of the most common misreadings, covered below.

Single-turn EV is the right objective surprisingly often. Most of a Yahtzee score comes from playing each turn well. The cross-turn corrections matter mainly for the upper bonus and for deciding when to scratch a category, which you handle with judgment on top of the numbers.

The dice math behind the count

Five dice rolled fresh produce 6^5 = 7776 ordered outcomes. Order does not matter for scoring, so those collapse into 252 distinct multisets. The number 252 is not arbitrary; it is the count of multisets of size 5 from 6 values:

\binom{6 + 5 - 1}{5} = \binom{10}{5} = 252

Here n = 6 is the number of faces and k = 5 is the number of dice. The binomial \binom{10}{5} counts the ways to choose 5 items from 6 types with repetition allowed. Each of those 252 multisets carries a weight: how many of the 7,776 ordered rolls map to it. A five-of-a-kind like 3-3-3-3-3 has weight 1. A hand with all distinct values but one pair, say 1-1-2-3-4, has weight \frac{5!}{2!} = 60. Those weights are the probabilities the search uses.

When you reroll only some dice, the same logic applies to the smaller set. Reroll three dice and there are \binom{3+3-1}{3} = \binom{5}{3} = 10 distinct outcomes for those three, weighted out of 6^3 = 216 ordered rolls.

Expectimax in one turn

Expectimax alternates two kinds of nodes. At a choice node you pick the keep-set that maximizes EV. At a chance node the dice roll, and you average over outcomes weighted by probability. The value of a keep at a choice node is:

V(\text{keep}) = \sum_{r} p(r) \cdot \max_{a} V(\text{keep} \cup r, a)

Read it this way. For a given keep-set you sum over every reroll result r of the dice you let go. Each result has probability p(r) (its weight divided by the ordered total). For each result you form the new hand and again take the best action a, which is either keep-and-reroll again or stop and score. At the final stage, with no rerolls left, the value is simply the best open category score for that hand.

With at most two rerolls, the tree is tiny. Each hand offers at most 2^5 = 32 keep-sets, each chance node spans at most 252 outcomes, and the recursion is only two levels deep. That is why the whole search finishes in milliseconds with no simulation and no sampling error.

Reproducing the demo

The demo button loads the field defaults: dice showing 1-2-3-4-6, two rerolls left, and every category open. Follow the reasoning the advisor formalizes.

  1. You are one pip away from a large straight (2-3-4-5-6) and also one pip away from a small straight already present in 1-2-3-4. The dice 1-2-3-4 alone are a small straight.
  2. Consider keeping 1-2-3-4 and rerolling the 6. A large straight needs a 5, which lands with probability \frac{1}{6} \approx 0.167 per reroll. Across two rerolls, chained through the tree, the chance of completing it is higher, and even a miss usually leaves the 30-point small straight.
  3. Consider keeping 2-3-4-6 and rerolling the 1. That does not help a straight and wastes the setup.
  4. Consider keeping all five. You lock in the 30-point small straight now but give up any reroll, so its EV equals a guaranteed 30, which the search compares against the higher average of chasing the large straight.

The advisor ranks keeping 1-2-3-4 at the top, reports the EV near 13.4 when it scores across all categories fairly (not just straights), and highlights those four dice. If you stopped right now it would name small straight as the best open category for 1-2-3-4-6, worth 30 points.

With the dice fixed at 1-2-3-4-6 and all categories open, keeping the four straight dice and rerolling one die gives the highest expected value; keeping all five locks in a 30-point small straight but forfeits the reroll, and keeping only the pairless run of three scores lower still.

Reading the ranked results

The output lists every keep option sorted by EV, with the best one highlighted, plus the best category to take if you stopped now. Two habits make the list trustworthy.

First, look at the gap, not just the order. If the top keep scores 13.4 and the second scores 13.1, those are practically equal and either is fine. If the top scores 13.4 and the second scores 9.0, the choice matters a lot.

Second, remember the number is an average over futures, not a promise. A keep worth 13.4 can still leave you with a poor hand on any single turn. The EV is the long-run rate at which that decision pays off.

Keeping the four-die run tops the ranking. Keeping all five locks a guaranteed value but forfeits both rerolls, so it falls below several rerolling options.

Common mistakes

The biggest one is ignoring the upper-section bonus. The advisor's EV does not include the 35-point bonus you earn for scoring at least 63 in the upper section (a rough average of three of each number). When you are chasing that bonus, favor the upper category even when the table shows a near-tie against a lower category.

Two hands that both show, say, three 5s might tie in raw EV between "fifteen in Fives" and "fifteen in Three of a Kind." The Fives entry advances the bonus; the Three of a Kind does not. Break near-ties toward the upper section when 63 is still in reach.

A second mistake is treating a second Yahtzee as a bonus. The advisor uses no joker rules. A second Yahtzee is scored as whatever the best open category pays, with no extra 100. If your house rules give a Yahtzee bonus, the EV understates those hands.

A third mistake is reading the "best category now" line as advice to stop. It only reports the best score if you stopped this instant. With rerolls left, the ranked keep list is the actual recommendation.

Related tools on this site

If exact game math interests you, several neighbors use the same style of complete enumeration. The Risk Battle Odds Calculator computes exact win probabilities from the attack and defense dice. The Catan Dice & Placement Odds ranks settlement spots by expected yield from the two-dice distribution. The Deck Odds Calculator gives exact draw probabilities using the same combinatorics as the 252 count here. For forced-outcome search rather than averaging, see the Chess Mate-in-N Solver. And for a cube decision driven by win rates, the Backgammon Doubling Cube Advisor shows the thresholds directly.

Frequently asked questions

Why 252 outcomes and not 7,776?

The 7,776 figure counts ordered rolls, treating a red 3 and a blue 3 as different. Scoring only cares about how many of each value you have, so those collapse to 252 distinct multisets. The search still respects the ordered counts as weights: a hand with more orderings is proportionally more likely.

Is the answer exact or simulated?

Exact. There is no random sampling. Every reroll branch is enumerated and every probability comes from the dice math itself, so two runs on the same position give identical numbers to the last decimal.

Does the EV include the 35-point upper bonus?

No. The EV is the points scored into the best open category for this hand alone. When you are close to the 63-point upper total, lean toward the upper category on near-ties even though the number does not show the bonus.

What happens with only one reroll left?

The tree shrinks to a single chance node. The advisor still enumerates every outcome of the dice you reroll, scores each resulting hand against the best open category, and averages. The recursion is one level instead of two.

Can it plan across multiple turns?

No. It optimizes the current hand in isolation. Cross-turn choices, such as which category to sacrifice on a bad roll, are left to you. The single-turn EV is still the correct core of almost every good decision.

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