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The situations where the strategy to maximize the average score differs are all after the first throw and are as follows: keep 44, 55, 66 rather than , keep 4, 5, or 6 rather than 11, keep 6 rather than Following the strategy to maximize the average score one will get a four-of-a-kind A player will keep a Yahtzee or Full House.
A Yahtzee will score 25 under the Joker rule, even though it is not strictly a full house. Keep any four-of-a-kind, three-of-a-kind, or pairs that are thrown and re-roll the others.
With two pairs keep both. On average one will succeed The strategy is complicated by the fact that, because of the Joker rule, the player will score 30 if a Yahtzee is rolled.
Clearly one keeps any Small Straight or Yahtzee that is thrown. After the first throw keep a run of 3 or 3 out of 4 e.
Otherwise, keep a 3 or 4 or both and a 2 or 5 if one also has a 3 or 4 e. Do not keep just 2, 5, or The only difference after the second throw is that one keeps , , and , and will try to throw a Yahtzee unless the other die is a 3 or 4.
Again the strategy is complicated by the fact that, because of the Joker rule, one will score 40 if one gets a Yahtzee. Clearly one will keep any Large Straight or Yahtzee that is thrown.
The best strategy is also to keep a four-of-a-kind and try to throw a Yahtzee, even after the first throw. The player should not keep a three-of-a-kind.
Keep a small straight or 4 out of 5 e. Otherwise one should simply keep any 2, 3, 4 or 5 just one of each , so that with one would keep A player should keep any Yahtzee, four-of-a-kind, three-of-a-kind or pair that is thrown and re-roll the others.
On average one will succeed 4. To get the maximum average score the strategy is straightforward. After the first throw the player will keep any 5s and 6s.
After the second throw one will keep any 4s, 5s, and 6s. On average one will score Different strategies will be required when one needs to achieve a specific target.
The strategy for maximising the expected score has been determined. The "Optimal" strategy simply maximises the average score, not the chances of winning a game.
There are two main reasons for this. Firstly, the Optimal strategy takes no account of any opponents. In normal gameplay a player will adjust their strategy depending on the scores of the other player or players.
Secondly, the Optimal strategy tends to give undue importance to Yahtzee bonuses. It is rare for a player without a Yahtzee bonus to beat a player with one.
So, in normal gameplay, a player's strategy is not significantly influenced by the value given for a Yahtzee bonus. Consider the situation where the Yahtzee bonus was worth a million rather than a hundred.
It would not influence normal gameplay where the objective is to score more than the opponent. It would, however, affect the "Optimal" strategy since scoring a million would have a dramatic effect on the average score.
The "Optimal" strategy would be dominated by the prospect of a Yahtzee bonus. Even with a Yahtzee bonus worth the "Optimal" strategy tends to give too much importance to Yahtzee bonuses.
Despite these limitations the "Optimal" strategy does provide a useful guide as to the best strategy, especially in the early rounds. The "Optimal" strategy for the first round is described in the next section.
After the first round, the number of different games rapidly makes detailed analysis difficult but, in the early rounds, players generally simply adapt the first round strategy based on the boxes used.
For instance if a player throws in the first round they will keep 66 but if the 6s box was used in the first round and they throw in the second round they will naturally keep 55 instead.
The following table shows the average score obtained using the Optimal strategy and the proportion of the time that zero is scored in a particular category: .
It is possible to calculate the maximum average score with different rules. When the rules are changed so that there is no Upper Section bonus the average score drops from This compares with an average bonus score of The average number of Yahtzees thrown in a game is 0.
In about 3. In about a quarter of these cases the Yahtzee box has a score 0 and so there is no Yahtzee bonus.
Although the average score is The 1st Percentile is i. Source:  . The strategy for the first round described here is based on that used by the "Optimal" strategy.
Sometimes there are two possible plays which are almost equally good. In these cases the simplest option is given, i. Although there are different ways of throwing 5 dice, because the order of the dice is not important there are only different combinations.
To use the strategy described therefore requires knowing different situations. The general principle after the first and second throws is to keep the largest number of similar dice and rethrow the rest.
When a player has a full house, keep the three-of-a-kind. With two pairs, keep the higher pair and rethrow the other three dice.
If a large straight is rolled, keep it. If a small straight is rolled, keep it and re-roll the fifth die. If all the dice are different and there is no straight, keep only the 5.
There are a number of exceptions to these rules. These exceptions are different for the second and third rolls.
So, before the second roll one will never keep two pairs, one will never keep a pair of 1s and one will never throw all the dice again.
So there are specific differences compared to the second roll. Now one will keep the Full House if the three-of-a-kind is 2 or 3 and keep two pairs when they are 1 and 2 or 1 and 3.
Keep all large and small straights. One will normally keep a pair of 1s although is still better and keep rather than 5 when all the dice are different.
The dice can be rolled a total of three times — the initial roll, plus two re-rolls of any or all dice. After rolling three times, the roll must be scored.
Any roll can be scored in any category at any time, even if the resulting score is zero. All upper section categories score a sum of the specified die face.
If the total score of all upper section categories is 63 or more, an extra bonus of 35 is added. Spelling Scramble. Mind Games for 2 Player.
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