Kniffel Yahtzee Kniffel / Yahtzee vereinfacht
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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: .
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 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.
In order to score 1, the player must throw 13 successive Yahtzees. The probability of getting a Yahtzee, if that is the sole objective, is 4.
The probability of scoring 1, if that was the sole objective is less than this since to score the maximum one 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 three-of-a-kind, four-of-a-kind and chance categories.
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, is considerably less. Using the strategy that maximizes the average score, the probability of scoring 1, is 1.
The highest possible score without a Yahtzee bonus is , which is achieved with scoring a five-of-a-kind as a Yahtzee, 84 in the Upper Section four-of-a-kind in each category , and 29 in each of 3 of a kind, 4 of a kind, and Chance.
The highest possible score without rolling a five-of-a-kind is 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.
A player must throw at least 8 Yahtzees in order to score Throwing 8 Yahtzees will score 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, 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. 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 A score of 18 will be achieved if is thrown each round and then, whatever is re-thrown, getting 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 the average score, is less likely than scoring the maximum of 1, but is theoretically possible.
Suppose that in the first two rounds after each roll. This would happen if a is thrown on the first roll and then, whatever dice are re-thrown, getting the same combination again; for instance, keeping 55, rethrowing the , and still getting The optimal strategy uses 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 results after each roll. Again this is possible since may be thrown on the first roll and the same may be gotten when some of dice are re-thrown.
The optimal strategy uses the four-of-a-kind 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 is rolled each time. Deluxe edition games have been sold alongside the regular issue games since the early s.
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.
Since the s, Travel Yahtzee has been sold in various forms as part of Milton Bradley's line of travel games. Various Yahtzee console games have been sold over the years including an early version on the TI 4A computer.
In , the game was first released to PC and Mac users by Atari. 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.
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 ,   as well as Google Play.
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.
The first was Triple Yahtzee, developed in There are a large number of versions of Yahtzee which can be played online or are available to download.
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: .
From Wikipedia, the free encyclopedia. This article is about the dice game. For the video game journalist, see Ben "Yahtzee" Croshaw.
This article may contain an excessive amount of intricate detail that may interest only a particular audience. Please help by spinning off or relocating any relevant information, and removing excessive detail that may be against Wikipedia's inclusion policy.
May Learn how and when to remove this template message. Although Yahtzee is a commercial game with its own components game set in upper image , the game can also be played with just five dice such as the dice in the lower image , some pencils or pens and sheets of paper, and the knowledge of the rules and scoring categories.
Main article: Games related to Yahtzee. This article appears to contain trivial, minor, or unrelated references to popular culture. Please reorganize this content to explain the subject's impact on popular culture, providing citations to reliable, secondary sources , rather than simply listing appearances.
Unsourced material may be challenged and removed. April Lowe Company". Archived from the original on August 8, Retrieved Milton Bradley Company.
Advantage Yahtzee. Las Vegas: Huntington Press. Epstein 28 December The Theory of Gambling and Statistical Logic. Academic Press. PC World.
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