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    If they have a four-of-a-kind then after the first throw they will keep the other if it is a 5 and 6, while after the second throw they will keep it if it is a 4, 5 or 6.

    So that with they keeps and will throw the 3. If they do not have a four-of-a-kind, the player should keep any three-of-a-kind or pair that they have and re-roll the other dice.

    With two pairs they will keep the higher pair. With no pair they will keep the highest die. Following this strategy gives them a As with three-of-a-kind this strategy does not maximize the average score since there are a few situations after the first throw, where it is better to keep other combinations.

    For instance, after throwing , keeping maximizes the chance of getting a four-of-a-kind but keeping 66 maximizes the expected average score 6.

    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 he 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. He will keep any four-of-a-kind, three-of-a-kind or pairs that are thrown and re-roll the others.

    With two pairs he will keep both. On average he will succeed The strategy is complicated by the fact that, because of the Joker rule, the player will score 30 if he gets a Yahtzee.

    Clearly he keeps any Small Straight or Yahtzee that he throws. After the first throw he will keep a run of 3 or 3 out of 4 e.

    Otherwise, he will keep a 3 or 4 or both and a 2 or 5 if he also has a 3 or 4 e. He will not keep just 2, 5 or The only difference after the second throw is that he 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, he will score 40 if he gets a Yahtzee.

    Clearly he will keep any Large Straight or Yahtzee that he throws. 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 he should simply keep any 2, 3, 4 or 5 just one of each , so that with he 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 he 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 he will keep any 4s, 5s, and 6s. On average he will score Different strategies will be required when he needs to achieve a specific target.

    The strategy for maximizing the expected score has been determined. The "Optimal" strategy simply maximises the average score.

    It does not maximise 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 players' 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: [6].

    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: [5] [7]. 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 he will 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 he will never keep two pairs, he will never keep a pair of 1s and he will never throw all the dice again.

    So there are specific differences compared to the second roll. Now he 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.

    He will Keep all large and small straights. He will Normally keep a pair of 1s although is still better and keep rather than 5 when all the dice are different.

    If he has any of the following he will put it in the appropriate Lower Section box: Yahtzee, full house, large straight, small straight.

    With a 4-of-a-kind or 3-of-a-kind he will put it in the appropriate Upper Section box. The only exception is that when he has a 3-of-a-kind where the sum of the dice is 25 or more he should use the 3-of-a-kind box.

    The 4-of-a-kind box is never used in the first round. In these situations there is no appealing option, the choice being between using an Upper Section box and using Chance.

    With two pairs follow the rule for the lower of the two pairs he has. The rule depends on whether his pair or smaller pair is low 1, 2 or 3 or high 4, 5 or 6 :.

    The best outcome in the first round is to throw a Yahtzee and put it in the Yahtzee box. This increases the final expected score from The second best outcome is to score 24 in the 6s box.

    This has a value of The worst first roll is which has a value of The worst third roll is which has a value of The highest possible score is 1, and involves throwing 13 Yahtzees, scoring 12 Yahtzee bonuses of points each, as well as in the other categories.

    The is achieved by scoring five aces 5 , five twos 10 , five threes 15 , five fours 20 , five fives 25 , five sixes 30 , the Upper Section bonus 35 , five sixes scored as Three-of-a-kind 30 , five sixes scored as a four-of-a-kind 30 , a full house 25 , a small straight 30 , a large straight 40 , a Yahtzee 50 , and five sixes scored as chance The score of 1, requires using the "Joker" rules for the full house, small straight, and large straight categories.

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