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 15566 in the first round, they will keep 66; however, if the 6s box was used in the first round and they throw 15566 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: The bonus is made in 68% of games, but in only 1% of games does the player get exactly 3-of-a-kind of each Upper Section category. In a further 5% of games, the player gets at least 3-of-a-kind in each of these categories and 4-of-a-kind in one or more of them. In the remaining 62% of games, there is at least one category which scores a pair or less but the bonus is achieved by scoring one or more 4-of-a-kind in the Upper Section. 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 254.59 to 237.84, a reduction of 16.75. This compares with an average bonus score of 23.84 and shows that changes to the strategy to improve the chance of getting the bonus are to the detriment of the other scores; the average score in the other boxes drops by 7.09. A Yahtzee is scored in about 34% of games while a Yahtzee bonus is scored in about 8% of games. A second Yahtzee bonus occurs about 1% of the time while a third one is scored in about 0.1% of games. The average number of Yahtzees thrown in a game is 0.46. This includes about 3% of games where a Yahtzee is thrown after 0 has been scored in the Yahtzee box and although it does score something, it is not scored as a Yahtzee or Yahtzee bonus. In about 3.7% of games a Yahtzee