Yahtzee

Yahtzee is a game played with five 6-sided dice. Players take turns rolling the dice, and trying to get certain types of rolls, each with an assigned point value, as summarized in the following table. Players are allowed a total of three rolls, with any subset of dice capable of being set aside at each roll. In addition to runs of a single number, other rolls include 3 of a kind (three of the same number), 4 of a kind (four of the same number), full house (two of one number and three of another), small straight (4 numbers in a row), large straight (5 numbers in a row), Yahtzee (five of the same number), and chance (any roll).

In addition to points scored by obtaining various rolls of the dice, 35 bonus points are awarded if the total score from the first six categories exceeds 62.

Woodward (2003) has solved the game completely by computing all possible outcomes and working out optimal playing strategies (Peterson 2003, Woodward 2003). T. Verhoeff and E. Scheffers have independently computed all the optimal strategies, and appear to arrive at the same results as Woodward.

In a variant of the game known as triple Yahtzee, players try to get each type of roll three times over the course of the game instead of just once, with point values for each roll being placed in a single, double, or triple column, whose values are multiplied by the stated weight when scores are totaled. The following tables summarizes the probability of obtaining various rolls. In this table, lower-value rolls are excluded from the results, so, for example, the probability of obtaining a three of a kind excludes rolls that are actually fours of a kind or Yahtzees. Similarly, the three of a kind probability excludes rolls that are full houses, and the two of a kind probability excludes rolls that are small straights.

type	1	2	3	overall
2 of a kind
3 of a kind
4 of a kind
full house
sm. straight
lg. straight
Yahtzee

Yahtzee was invented by a wealthy Canadian couple to play aboard their yacht. In 1956, the couple approached businessman Edwin S. Lowe, who had made a fortune selling Bingo games in the 1920s, who bought the rights and eventually changed the name of the "yacht game" to "Yahtzee." Despite attempts to advertise the game, it did not sell very well that first year. However, after Lowe started throwing Yahtzee parties, the game took off thanks to word of mouth and has been widely played ever since (Hasbro). Hasbro, the game's current manufacturer, claims that as many as 100 million people worldwide play the game regularly (Peterson 2003).