Yahtzee
Present Yahtzee logo

Players

1+

Age range

8+

Playing time

30 minutes

Random chance

High

Skill(s) required

Luck, probability, strategy

Yahtzee is a dice game made by Milton Bradley (now owned by Hasbro), which was first marketed as "Yatzie" by National Association Service of Toledo, Ohio, in the early 1940s. Yatzie was included in a game set called "LUCK  15 Grand Dice Games". It was marketed under the name of Yahtzee by game entrepreneur Edwin S. Lowe in 1956. The game is a development of earlier dice games such as Poker Dice, Yacht and Generala. It is also similar to Yatzy, which is popular in Scandinavia.
The object of the game is to score points by rolling five dice to make certain combinations. The dice can be rolled up to three times in a turn to try to make various scoring combinations. A game consists of thirteen rounds. After each round the player chooses which scoring category is to be used for that round. Once a category has been used in the game, it cannot be used again. The scoring categories have varying point values, some of which are fixed values and others where the score depends on the value of the dice. A Yahtzee is fiveofakind and scores 50 points; the highest of any category. The winner is the player who scores the most points.
Yahtzee was marketed by the E.S. Lowe Company from 1956 till 1973. In 1973, the Milton Bradley Company purchased the E.S. Lowe Company and assumed the rights to produce and sell Yahtzee. During Lowe's ownership over 40 million Yahtzee games were sold in America and around the globe. The game has maintained its popularity. According to current owner Hasbro, 50 million Yahtzee games are sold each year. A classic edition is currently being marketed by Winning Moves.
Contents

History 1

Rules 2

Upper section 2.1

Lower section 2.2

Yahtzee bonuses and Joker rules 2.3

Forced Joker rule 2.3.1

Free choice Joker rule 2.3.2

Original Joker rule 2.3.3

Winner 2.4

Example game 3

Strategy 4

Completing a Yahtzee 4.1

Specific Yahtzee 4.1.1

Any Yahtzee 4.1.2

Final round 4.2

Upper Section box 4.2.1

ThreeOfAKind 4.2.2

FourOfAKind 4.2.3

Full House 4.2.4

Small Straight 4.2.5

Large Straight 4.2.6

Yahtzee 4.2.7

Chance 4.2.8

Optimal strategy 4.3

First round 4.4

Second and third roll choices 4.4.1

Category choice 4.4.2

Best and worst outcome 4.4.3

Extreme scores 5

Maximum score 5.1

Other high scores 5.2

Minimum score 5.3

Commercial versions 6

Deluxe and collector editions 6.1

Travel Yahtzee 6.2

Electronic versions 6.3

Related games 6.4

Online versions 7

Similar games 8

Cultural references 9

References 10

External links 11
History
The Original 1956 Yahtzee Board Game
The overall concept of Yahtzee traces its roots to a number of traditional dice games. Among these is the Puerto Rican game Generala, and the English games of Poker Dice and Cheerio. Another game, Yap, shows close similarities to Yahtzee; this game was copyrighted by Robert Cissne in 1952.
The most important predecessor of Yahtzee is the dice game named "Yacht", which is an English cousin of Generala and dates back to at least 1938.^{[1]} Wood^{[1]} classifies Yacht, and a similar threedice game called Crag, as sequence dice games. Yahtzee is similar to Yacht in both name and content. Although Yahtzee is clearly derived from Yacht, it differed from it in a number of significant ways:

It introduced the upper section bonus.

It included the threeofakind category.

In Yacht both straights are a sequence of five ("Big Straight" is 23456, "Little Straight" is 12345). Yahtzee introduced the sequence of four straight (the "Small Straight").

It introduced Yahtzee bonuses and the Joker rule.
There were also a number of scoring differences. The presentday commercial Yahtzee began when toy and game entrepreneur Edwin S. Lowe filed Yahtzee as a trademark with the U.S. Patent Office on April 19, 1956. The first commercial usage of the name Yahtzee was a few weeks earlier on April 3. Lowe classified his product as a "Poker Dice Game."
According to Hasbro, the game was invented by an anonymous Canadian couple, who called it "The Yacht Game" because they played it on their yacht with their friends.^{[2]} Later they asked Lowe if he would make up some sets to be given as gifts to their friends who enjoyed the game. Lowe perceived the possibility of marketing the game, and acquired the rights to the game from the couple in exchange for 1,000 gift sets. This story is expanded by Lowe in the 1973 book, A Toy is Born by Marvin Kaye. According to Lowe, the game did not initially do well commercially, since the rules and appeal were not easily conveyed in an advertisement. Eventually, he had the idea of organizing Yahtzee parties at which people could play the game and thereby gain a firsthand appreciation of it. The idea was successful, and enthusiasts quickly popularized the game through word of mouth.
The E.S. Lowe Company sold Yahtzee from 1956 to 1973. During Lowe's ownership, a number of changes were made to the game's packaging, contents, and appearance. Between 1956 and 1961, the game's advertising slogan was changed from "The Game That Makes You Think While Having Fun" to "The Fun Game That Makes Thinking Fun!"
The game and its contents were copyrighted by Lowe in 1956, 1961, 1967, and 1972. In 1973, Milton Bradley purchased the E.S. Lowe Company and assumed the rights to produce and sell Yahtzee. During Lowe's ownership over 40 million Yahtzee games were sold in America and around the globe. The game has maintained its popularity. According to current owner Hasbro, 50 million Yahtzee games are sold each year.
Over time, the Yahtzee logo has taken several forms. The original version of the logo was used throughout the entire period that the game was produced solely by the Lowe company. After 1973, the logo changed various times. This logo is found on the scorecards and the game boxes.
Rules
Logo of the game as originally published in 1954. It includes a caricature of E. S. Lowe, the person responsible for the game's production.
The game consists of a number of rounds. In each round, a player gets three rolls of the dice, although they can choose to end their turn after one or two rolls. After the first roll the player can save any dice they want and reroll the other dice. This procedure is repeated after the second roll. The player has complete choice as to which dice to roll. They can reroll a dice for the third roll that was not rolled on the second roll.^{[3]}
The Yahtzee scorecard contains 13 different category boxes and in each round, after the third roll, the player must choose one of these categories. The score entered in the box depends on how well the five dice match the scoring rule for the category. Details of the scoring rules for each category are given below. As an example, one of the categories is called ThreeofaKind. The scoring rule for this category means that a player only scores if at least three of the five dice are the same value. The game is completed after 13 rounds by each player, with each of the 13 boxes filled. The total score is calculated by summing all thirteen boxes, together with any bonuses.^{[3]}
The Yahtzee scorecard contains 13 scoring boxes divided between two sections: the upper section and the lower section.^{[3]}
Upper section
In the upper section there are six boxes. The score in each of these boxes is determined by adding the total number of dice matching that box.
Category

Description

Score

Example

Aces

Any combination

The sum of dice with the number 1

scores 3

Twos

Any combination

The sum of dice with the number 2

scores 6

Threes

Any combination

The sum of dice with the number 3

scores 12

Fours

Any combination

The sum of dice with the number 4

scores 8

Fives

Any combination

The sum of dice with the number 5

scores 0

Sixes

Any combination

The sum of dice with the number 6

scores 18

If a player scores a total of 63 or more points in these six boxes, a bonus of 35 is added to the upper section score. Although 63 points corresponds to scoring exactly threeofakind for each of the six boxes, a common way to get the bonus is by scoring fourofakind for some numbers so that fewer of other numbers are needed. A player can earn the bonus even if they score a "0" in an upper section box.^{[3]}
In order to gauge how well a player is doing in the upper section, they often refer to being "up" or "down" compared to the average of three required for each box. So that if a player scores four "sixes" they will be "6 up"; while if they then score just two "twos" they will then be only "4 up". Similarly, if a player starts with two "twos" they will be "2 down".^{[3]}
Lower section
The lower section contains a number of pokerthemed categories with specific point values:
Category

Description

Score

Example

ThreeOfAKind

At least three dice the same

Sum of all dice

scores 17

FourOfAKind

At least four dice the same

Sum of all dice

scores 24

Full House

Three of one number and two of another

25

scores 25

Small Straight

Four sequential dice
(1234, 2345, or 3456)

30

scores 30

Large Straight

Five sequential dice
(12345 or 23456)

40

scores 40

Yahtzee

All five dice the same

50

scores 50

Chance

Any combination

Sum of all dice

scores 13

Some players count a Yahtzee as being a valid Full House. However the official rule is that a Full House is "three of one number and two of another".^{[3]}
Small Straight and Large Straight are sometimes called by different names, either Low Straight and High Straight, Short Straight and Long Straight or Little Straight and Big Straight.^{[3]}
If a category is chosen but the dice do not match the requirements of the category the player scores 0 in that category. Some combinations offer the player a choice as to which category to score them under; e.g., a full house could be scored in the Full House, the ThreeOfAKind, or the Chance categories. The Chance category is often used for a turn that will not score well in any other category.^{[3]}
Yahtzee bonuses and Joker rules
A Yahtzee occurs when all five dice are the same. If a player throws a Yahtzee but the Yahtzee category has already been used, special rules apply.^{[3]}
If the player throws a Yahtzee and has already filled the Yahtzee box with a score of 50, they score a Yahtzee bonus and get an extra 100 points. However, if they throw a Yahtzee and have filled the Yahtzee category with a score of 0, they do not get a Yahtzee bonus.^{[3]}
In either case they then select a category, as usual. Scoring is the same as normal except that, if the Upper Section box corresponding to the Yahtzee has been used, the Full House, Small Straight and Large Straight categories can be used to score 25, 30 or 40 (respectively) even though the dice do not meet the normal requirement for those categories. In this case the Yahtzee is said to act as a "Joker".^{[3]}
There are two alternative versions of the Joker rule used.^{[3]}
Forced Joker rule
In the official rules^{[3]} the player must act in the following way.

If the corresponding Upper Section box is unused then that category must be used.

If the corresponding Upper Section box has been used already, a Lower Section box must be used. The Yahtzee acts as a Joker so that the Full House, Small Straight and Large Straight categories can be used to score 25, 30 or 40 (respectively).

If the corresponding Upper Section box and all Lower Section boxes have been used an unused Upper Section box must be used, scoring 0.
Free choice Joker rule
In the simpler, alternative version of the Joker rule^{[4]}^{[5]} the player retains the free choice as to which category to use, but the Yahtzee can only be used as a Joker if the corresponding Upper Section box has been used. If the corresponding Upper Section box is unused the Yahtzee would score 0 if the Full House, Small Straight or Large Straight categories were chosen.^{[3]}
Original Joker rule
The original game rules released in 1956 contain a difference from the above rules. The booklet stated that additional Yahtzees must be used as Jokers in the Lower Section and did not allow for their use in the Upper Section. This rule was changed when the game was recopyrighted in 1961.^{[3]}
Winner
The winner is the player with the highest total. The rules do not specify what happens in the event of a tie.^{[3]}
Example game
Strategy
A detailed discussion of the optimal strategy is beyond the scope of this article but certain aspects can be considered here.
Completing a Yahtzee
It is important to understand the probability of completing a Yahtzee. This depends on whether a particular Yahtzee is required or any Yahtzee.
Specific Yahtzee
The probability of completing a specific Yahtzee is shown in the following table.
Number Already
Rolled

Before First
Roll

Before Second
Roll

Before Third
Roll

4



30.56%

16.67%

3



9.34%

2.78%

2



2.85%

0.46%

1



0.87%

0.08%

0

1.33%

0.27%

0.01%

Any Yahtzee
The probability of completing any Yahtzee is shown in the following table. Keep any Yahtzee, fourofakind, threeofakind or pair that you throw and reroll the others. With two pairs you can choose either one.
Most Already
Rolled

Before First
Roll

Before Second
Roll

Before Third
Roll

4



30.56%

16.67%

3



9.34%

2.78%

2



2.91%

0.46%

1



1.26%

0.08%

0

4.60%





So if you throw a threeofakind on the first roll, the chance of completing a Yahtzee has increased from 4.60% to 9.34%.
Final round
This section describes the last round strategy where there is no possibility of a Yahtzee bonus (i.e. there is a score of 0 in the Yahtzee box). These are the simplest situations to analyse, although even at this late stage the strategy may vary depending on the game situation.
Upper Section box
If the final box is in the upper section, the strategy is simple. Keep any of the number required and reroll the others. On average you will get 2.11 of the number you require, scoring from 2.11 (for 1s) to 12.64 (for 6s). The distribution is as follows:
Total

Probability

At least 1

93.51%

At least 2

69.88%

At least 3

35.48%

At least 4

10.44%

5

1.33%

ThreeOfAKind
There are slightly different strategies depending on whether you are simply just trying to get a threeofakind or you are trying to maximise your average score. Different strategies will also be required should you need to achieve a specific target.
The strategy to maximise your chance of getting a threeofakind involves keeping any threeofakind that you have. In you have a threeofakind then after the first throw keep any other 5s and 6s, while after the second throw keep any other 4s, 5s, and 6s. So that with 22225 you keep 2225 and throw the other 2. If you do not have a threeofakind keep any pair that you have and reroll the other dice. With two pairs keep the higher pair. With no pair keep the highest dice. Following this strategy gives you a 74.32% chance of getting a threeofakind.
This strategy does not maximise 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 11166, keeping 11166 maximises the chance of getting a threeofakind (a guaranteed score of 15) but keeping 66 maximises the expected average score (17.54), although it introduces a significant change of scoring nothing. The situations where the strategy to maximise the average score differs are all after the first throw and are as follows: keep 33, 44, 55, 66 rather than 111 (or 11155, 11166), keep 4, 5 or 6 rather than 11, keep 5 or 6 rather than 22, keep 336 rather than 33. So that with 22456 keep 6 rather than 22 and with 33456 keep 336 not 33. Following the strategy to maximise the average score you will get a threeofakind 71.51% of the time and score an average of 15.19.
FourOfAKind
As with threeofakind there are slightly different strategies depending on whether you are simply trying to get a fourofakind or you are trying to maximise your average score. Different strategies will also be required should you need to achieve a specific target.
The strategy to maximise your chance of getting a fourofakind involves keeping any fourofakind that you have. In you have a fourofakind then after the first throw keep the other if it is a 5 and 6, while after the second throw keep it if it is a 4, 5 or 6. So that with 22223 you keep 2222 and throw the 3. If you do not have a fourofakind keep any threeofakind or pair that you have and reroll the other dice. With two pairs keep the higher pair. With no pair keep the highest dice. Following this strategy gives you a 29.08% chance of getting a fourofakind.
As with threeofakind this strategy does not maximise 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 11166, keeping 111 maximises the chance of getting a fourofakind but keeping 66 maximises the expected average score (6.20 rather than 4.18). The situations where the strategy to maximise the average score differs are all after the first throw and are as follows: keep 44, 55, 66 rather than 111, keep 4, 5 or 6 rather than 11, keep 6 rather than 22. Following the strategy to maximise the average score you will get a fourofakind 27.74% of the time and score an average of 5.61.
Full House
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 fourofakind, threeofakind or pairs that you throw and reroll the others. With two pairs keep both. On average you will succeed 36.61% of the time and score an average of 9.15.
Small Straight
The strategy is complicated by the fact that, because of the Joker rule, you will score 30 if you get a Yahtzee. Clearly you keep any Small Straight or Yahtzee that you throw. After the first throw keep a run of 3 or 3 out of 4 (e.g. 124) if you have one. Otherwise keep a 3 or 4 (or both) and a 2 or 5 if you also have a 3 or 4 (e.g. with 11245 keep 24 but with 11255 keep nothing). Do not keep just 2, 5 or 25. The only difference after the second throw is that you keep 1111, 2222, 5555 and 6666 and try to throw a Yahtzee unless the other dice is a 3 or 4.
On average you will succeed 61.60% of the time and score an average of 18.48.
Large Straight
Again the strategy is complicated by the fact that, because of the Joker rule, you will score 40 if you get a Yahtzee. Clearly you keep any Large Straight or Yahtzee that you throw. The best strategy is also to keep a fourofakind and try to throw a Yahtzee, even after the first throw. Do not keep a threeofakind. Keep a small straight or 4 out of 5 (e.g. 1235) if you have one. Otherwise simply keep any 2, 3, 4 or 5 (just one of each), so that with 12256 you would keep 25.
On average you will succeed 26.53% of the time and score an average of 10.61.
Yahtzee
Keep any Yahtzee, fourofakind, threeofakind or pair that you throw and reroll the others. On average you will succeed 4.60% of the time and score an average of 2.30.
Chance
To get the maximum average score the strategy is straightforward. After the first throw keep any 5s and 6s. After the second throw keep any 4s, 5s, and 6s. On average you will score 23.33. Different strategies will be required when you need to achieve a specific target.
Optimal strategy
The strategy for maximizing the expected score has been determined.^{[4]} The average score when using this strategy is 254.59.^{[5]} This average is based on calculations using the "free choice Joker rule".
It is important to note that the "Optimal" strategy simply maximises the average score. It does not maximise your 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 100 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 but 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:^{[5]}
Category

Average
score

Zero
score

Aces

1.88

10.84%

Twos

5.28

1.80%

Threes

8.57

0.95%

Fours

12.16

0.60%

Fives

15.69

0.50%

Sixes

19.19

0.53%

Bonus

23.84

31.88%

ThreeofaKind

21.66

3.26%

FourofaKind

13.10

36.34%

Full House

22.59

9.63%

Small Straight

29.46

1.80%

Large Straight

32.71

18.22%

Yahtzee

16.87

66.26%

Chance

22.01



Yahtzee Bonus

9.58

91.76%

TOTAL

254.59



The bonus is made in 68% of games but in only 1% of games does the player get exactly 3ofakind of each Upper Section category. In a further 5% of games the player get at least 3ofakind in each of these categories and 4ofakind in one or more then. 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 4ofakind 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.^{[5]} 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 is used as a Joker and scored on Full House, Small Straight or Large Straight.^{[5]} In most of these cases the Yahtzee box has a score of 50 and so the Yahtzee also counts as a Yahtzee bonus. 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 254.59 the Median score is 248^{[5]} The Mode is 266 which occurs 1.26% of the time. The 1st Percentile is 151 (i.e. 1% of scores are 151 or less) while the 99th Percentile is 473 (i.e. 99% of scores are 473 or less).
First round
Source:^{[4]}^{[6]}
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.e. the one that follows the general rules.
Although there are 7776 different ways of throwing 5 dice, because the order of the dice is not important there are only 252 different combinations. To use the strategy described therefore requires knowing 756 different situations.
Second and third roll choices
The general principle after the first and second throws is to keep the largest number of similar dice and rethrow the rest. When you have a full house keep the threeofakind. When you have two pairs keep the higher pair and rethrow the other three dice. If you have a large straight, keep it. If you have a small straight, keep it and reroll the fifth dice. If all the dice are different and there is no straight, keep just the 5. There are a number of exceptions to these rules. These exceptions are different for the second and third rolls.
Exceptions before the second roll:

with a full house containing three 1s, keep all the dice and put in the Full House box.

with 12344 keep the pair not the small straight.

with a 3456 and a pair keep the pair not the small straight.

with a pair of 1s keep, in order of preference, 345, 5, 4 or 6, not the pair of 1s.
So, before the second roll you never keep two pairs, never keep a pair of 1s and never throw all the dice again.
Exceptions before the third roll:

with a full house containing three 1s, three 2s or three 3s, keep all the dice and put in the Full House box.

with a pair of 1s and a pair of 2s or with a pair of 1s and a pair of 3s, keep both pairs.

with 11345 keep 345 not the pair.

with 12456 keep 456 not just the 5.
So there are specific differences compared to the second roll. Now keep the Full House if the threeofakind is 2 or 3 and keep two pairs when they are 1 and 2 or 1 and 3. Keep all large and small straights. Normally keep a pair of 1s (although 345 is still better) and keep 456 rather than 5 when all the dice are different.
Category choice
In about 80% of games you will either have at least a 3ofakind or a straight. If you have any of the following put it in the appropriate Lower Section box: Yahtzee, full house, large straight, small straight. With a 4ofakind or 3ofakind put it in the appropriate Upper Section box. The only exception is that when you have a 3ofakind where the sum of the dice is 25 or more you should use the 3ofakind box. The 4ofakind box is never used in the first round.
In the remaining 20% of cases you will have a poor hand: two pairs, one pair or all dice different. 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 you have. The rule depends on whether your pair (or smaller pair) is low (1, 2 or 3) or high (4, 5 or 6):

with a low pair use the appropriate Upper Section box (1s, 2s or 3s) unless the sum of the dice is 22 or more in which case use the Chance box.

with a high pair use the 1s box unless the sum of the dice is 20 or more in which case use the Chance box. The only exception is that with 23446 you should use the Chance box even though it only scores 19.

with all dice different use the 1s box.
Best and worst outcome
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 254.59 to 320.84.^{[5]} Putting a Yahtzee of 6s in the 6s box results in an expected score of just 279.48. The second best outcome is to score 24 in the 6s box. This has a value of 268.23.^{[5]} Putting 56666 in the 4ofakind box has a value of 264.54.
The worst first roll is 11236 which has a value of 249.83.^{[5]} After the second roll the worst position is 12356 which then has a value of 244.91. The worst third roll is 23446 which has a value of 238.96 when put it in the Chance box.^{[5]}
Extreme scores
Maximum score
The highest possible score is 1,575 and involves throwing 13 Yahtzees, scoring 12 Yahtzee bonuses of 100 points each, as well as 375 in the other categories.
The 375 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 Threeofakind (30), five sixes scored as a Fourofakind (30), a Full House (25), a Small Straight (30), a Large Straight (40), a Yahtzee (50), and five sixes scored as Chance (30). The score of 1,575 requires using the "Joker" rules for the Full House, Small Straight and Large Straight categories.
In order to score 1,575 the player must throw 13 successive Yahtzees. The probability of a getting a Yahtzee, if that is your sole objective, is 4.60% and so the probability of a player rolling 13 Yahtzees in a row (if that was their sole objective) is 4.60% raised to the 13th power or 4.16e18. The probability of scoring 1,575 (if that was your sole objective) is less than this since to score the maximum you would need at least one of each number (after the first round) to fill the Upper Section and a further three 6 Yahtzees to score 30 in the Threeofakind, Fourofakind and Chance categories. In about 18% of Yahtzees the player has some choice over the dice to keep. This most frequently happens when a player rolls two pairs. This means that a player can slightly increase the chance of getting a 6 Yahtzee without reducing the chance of getting a Yahtzee.
In normal gameplay the probability of scoring 1,575 is considerably less. Using the strategy that maximizes the average score, the probability of scoring 1,575 is 1.38e20.
Other high scores
For a player following the strategy which maximises the chance of getting a Yahtzee, the probability of scoring a particular number of Yahtzees follows a binomial distribution and is shown in the following table.
Yahtzees
Rolled

Probability

Chance

1 or more

45.80%

1 in 2.18

2 or more

11.81%

1 in 8.47

3 or more

1.97%

1 in 50.7

4 or more

0.210%

1 in 436

5 or more

1.82e4

1 in 5,140

6 or more

1.17e5

1 in 81,200

7 or more

5.66e7

1 in 1,700,000

8 or more

2.05e8

1 in 47,500,000

A player must throw at least 8 Yahtzees in order to score 1000. Throwing 8 Yahtzees will score 750 for the Yahtzee and Yahtzee bonuses. The 7 Yahtzees after the first throw will generally score highly, including the opportunity of using the Joker rule to score in, for instance, the Large Straight category. This means that a player throwing 8 Yahtzees will often be able to score 1,000.
In normal gameplay the probability of scoring this number of Yahtzees is considerably less. The following table shows the probability of getting certain scores using the "Optimal" strategy.
Score

Probability

Chance

400 or more

3.84%

1 in 25.5

500 or more

0.719%

1 in 139

600 or more

8.96e4

1 in 1,120

700 or more

7.50e5

1 in 13,300

800 or more

4.36e6

1 in 229,000

900 or more

1.83e7

1 in 5,480,000

1000 or more

5.58e9

1 in 179,000,000

Minimum score
The lowest possible score is 5. The Chance box always scores the sum of the dice, so that a minimum of 5 must be scored in that category.
The strategy that works to maximize the average expected score will under worst case conditions score a minimum of 12 points, but cautious play will guarantee a minimum score of 18.^{[7]}
You will get a score of 18 if you throw 11223 each round and then, whatever you rethrow, get 11223 again after the second and third throws, scoring 2 in the 1s box, 4 in the 2s box, 3 in the 3s box and 9 for the Chance category.
Scoring the minimum of 12, when attempting to maximise your average score, is less likely than scoring the maximum of 1,575 but is theoretically possible. Suppose that in the first two rounds you have 12355 after each roll. This would happen if you throw 12355 on the first roll and then, whatever dice you rethrow, getting the same combination again. For instance, you keep 55, rethrow the 123, and still get 12355. The optimal strategy will use the 1s box in the first round, scoring 1, and the 2s box in the second round, scoring 2. Now suppose that for the next 3 rounds you have 11226 after each roll. Again this is possible since you may throw 11226 on the first roll and then get the same when you rethrow some of dice. The optimal strategy will use the fourofakind and Yahtzees boxes in the third and fourth rounds and then the 3s box in the fifth round, scoring 0 each time. For the last 8 rounds suppose you have 11223 after each roll. You will score 0 in all the remaining categories except Chance where you will score 9, giving a total of 12.
Commercial versions
Deluxe and collector editions
Deluxe edition games have been sold alongside the regular issue games since the early 1960s. They all contain components that are more luxurious than standard game parts. In recent years, a number of collector issue Yahtzees have been sold as well. Some of these collector issues have dice that replace the pips with certain symbols connected to a theme, but still correspond to the numbers one to six.
Travel Yahtzee
1986 Travel Yahtzee
Since the 1970s, Travel Yahtzee has been sold in various forms as part of Milton Bradley's line of travel games.

In the past, the travel form of Yahtzee was composed of the five dice contained in a special apparatus (see image at the right). "Rolling" the dice is initiated by turning the apparatus over, running it on one hand and turning the apparatus upright to see the faces of the dice. Dice to be saved for the next roll are locked by snapping the reverse of the apparatus.

Currently, a zipup cloth deluxe folio edition is sold. It consists of the set of dice, the cup, the scorecards, and a tray for the dice to roll on, which includes a holder for the dice to be saved for the next roll. The cup has an oblong lip for easy storage.

There also exists a red circular travel edition, which includes the dice and scorecards, as well as a collapsible cup and detachable dice holder. The bottom half of the case acts as a dice tray. It should be noted that this one uses a different logo from the standard one (the name of the game in uppercase with an exclamation point in front of a green oval) and it is sold outside of the United States.^{[8]}

There are also miniature versions of the game, sold in sets contained either in cups that act as keychains (by the company Basic Fun) ^{[9]} or in specially shaped pens (by the company Stylus).^{[10]}
Electronic versions
Various Yahtzee console games have been sold over the years including an early version on the TI99 4A computer. In 1996, the game was first released to PC and Mac users by Atari. The Ultimate Yahtzee CDROM game contained standard Yahtzee as well as other varieties. Later, GameHouse also released an authorized special version of the game for Windows users.
There are also several electronic versions of the game such as a handheld LCD version, and a cell phone version called Yahtzee Deluxe, which feature the original rules along with Duplicate and Rainbow modes, as well as independently produced versions for the Palm OS and Pocket PC and several cellphone models. The version for the Nintendo Game Boy was licensed from Hasbro and was produced by DSI Games and Black Lantern Studios Inc. It was sold in a three pack that included Life, Payday, and Yahtzee. The game has also been released for the iPod, iPod touch and the iPhone, to be purchased through the iTunes Store,^{[11]}^{[12]} as well as Google Play. Both the iOS App and the Android App are published by Scopely. Yahtzee is available on the Xbox 360 in the Family Game Night game by Hasbro. Pogo.com released a version in 2009, and the game is also available on the Pogo Facebook site as well.
Related games
A number of related games under the Yahtzee brand have been produced. They all commonly use dice as the primary tool for gameplay, but all differ generally. As Yahtzee itself has been sold since 1954, the variants released over the years are more recent in comparison, with the oldest one, Triple Yahtzee, developed in 1972, eighteen years after the introduction of the parent game. In addition, the 1970s television game show SpinOff was based on Yahtzee. Another, similarly shortlived, TV game show adaptation, Yahtzee, was syndicated to local stations during 1988 season.
Online versions
There are a large number of versions of Yahtzee which can played online or are available to download.

"Open Yahtzee" is an crossplatform opensource (free) version of Yahtzee for one player.^{[13]}
Similar games
There are a large number of related games. Yatzy is one of the most popular. Yatzy rules and scoring categories are somewhat different from Yahtzee:^{[14]}

The bonus for reaching 63 or more points in the Upper Section is normally 50 points.

There are two extra boxes for the "One Pair" and "Two Pair" categories. These score the total of the pair(s) involved. For instance, 55441 will score 10 points in the One Pair box and 18 in the Two Pair box. In the Two Pair category, the pairs must be different.

The ThreeofaKind and FourofaKind categories are scored using the total of the needed number of samefaced dice. For instance, 55556 will score 20 points in FourofaKind and 15 points in ThreeofaKind. This scoring rule is the same as the FourofaKind scoring used in Yacht.

Full House scores the total of all dice, as in Yacht.

Small Straight and Large Straight have the same definitions as in Yacht. A Small Straight is 12345 and scores 15 points (the total of the dice faces); the Large Straight is 23456 and scores 20 points (again the combined value of the dice).

There are no Yahtzee bonuses or Joker rule. Yahtzees can be scored in other categories under the normal scoring rules. For example, a Yahtzee of sixes will score 12 in the One Pair box, 18 in ThreeofaKind, 24 in FourofaKind, 30 in Sixes or Chance and 0 in Two Pair, Full House, Small Straight and Large Straight boxes.
Other related games include:

Balut is the name of a Danish dice game played by expatriates in many countries all over the world. The name of the game is taken from balut eggs.

Kismet has dice with multiple colors. Both numbers and colors are taken into account when scoring.
Cultural references

Ben "Yahtzee" Croshaw is a British video games critic currently based in Australia who takes his pseudonym from the game.^{[15]}

Yahtzee has been cited as a precipitating factor in a case of domestic violence ^{[16]} and as the motive for at least one murder.^{[17]}

The word "Yahtzee" has been used to exclaim excitement when a University of Kentucky football recruit commits to attending the school.^{[18]}

The tennis player Andy Roddick, after winning a match in which events seemed to go his way, said, in an interview, that everything was "coming up Yahtzee".
References

^ ^{a} ^{b} Wood, Clement and Goddard, Gloria, The Complete Book of Games, Halcyon House, NY, 1938

^ "The History of YAHTZEE". web.archive.org. Retrieved 20150923.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} "Yahtzee Instructions – 1 or More Players" (PDF). Milton Bradley Company. 1996. Retrieved 20150923.

^ ^{a} ^{b} ^{c} Vancura, Olaf (2001). Advantage Yahtzee. Las Vegas: Huntington Press.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} Verhoeff, Tom (1999), Optimal Solitaire Yahtzee Strategies (PDF), retrieved 20140905

^ https://books.google.co.uk/books?id=g5YWIpHTTW8C&pg=PA417 The Theory of Gambling and Statistical Logic (pages 417 and 418)

^ "Optimal Solitaire Yahtzee Player: Trivia". wwwset.win.tue.nl. Retrieved 20150923.

^ "Image: 516X0Z50P4L._AA280_.jpg, (280 × 280 px)". ecx.imagesamazon.com. Retrieved 20150923.

^ "Image: yahtzee.jpg, (240 × 240 px)". ultimatekeychains.com. Retrieved 20150923.

^ http://ecx.imagesamazon.com/images/I/41qTHiYkvL._SS500_.jpg

^ Horwitz, Jeremy (February 18, 2008). "Electronic Arts Yahtzee (Review by iLounge Accessory Reviews)". Retrieved 20080221.

^ Cohen, Peter (February 16, 2008). "Yahtzee Comes to iPod". PC World (

^ "About  Open Yahtzee". openyahtzee.org. Retrieved 20150923.

^ Yahtzee's equivalent articles in Danish, Finnish, Norwegian, and Swedish

^ "Zero Punctuation Video Gallery  The Escapist". escapistmagazine.com. Retrieved 20150923.

^ Hoffer, Steven (28 September 2011). "Florida Man Allegedly Chokes Wife Over Yahtzee". Huffington Post.

^ "Dispute over Yahtzee game may be motive for mother's murder  kare11.com". kare11.com. Retrieved 20150923.

^ "Recruiting  UK  The CourierJournal  courierjournal.com". courierjournal.com. Retrieved 20150923.
External links

Comprehensive information on rules, gameplay and strategy

PYahtzee, Solution, Variants, Statistics, Tactics

Yahtzee Online Club. Play singleplayer and multiplayer Yahtzee with friends


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