Kniffel Chance

Kniffel Chance Kombinationen

(alle Augenzahlen zählen; maximal 30 Punkte) Die mittlere Punktzahl für eine. Kniffel oder Yahtzee ist ein Würfelspiel mit fünf Würfeln, einem Würfelbecher und einem Teil mit den sogenannten Figuren sowie die Einführung der „Chance“, bei der beliebige Wurfkombinationen addiert und eingetragen werden können. Chance - hier muss nichts zusammenpassen. Alle Augen werden zusammengezählt. Video: Kniffel LIVE für Android. Chance: Gibt dem Spieler die Möglichkeit, einen miss- glückten Wurf, der keine Bedingung erfüllt, trotzdem als Punkte in seinem „Chancefeld. Bei Kniffel wird die untere Tabelle mit dem Feld „Chance“ komplettiert. Ein Spieler, der keines.

Kniffel Chance

Ich kenne jemanden, der öfter einen Kniffel hatte. In fast jedem Spiel. Ich finde das Spiel frustrierend, haha ich habe da einfach kein Glück. Lässt sich eine gewürfelte Kombination in keines der ersten 12 Felder eintragen, so hat der Spieler einmalig durch das Feld "Chance". Die Kniffel, Yahtzee oder Würfelpoker mit Regeln und Spielablauf. für einen "​billigen Wurf" das Feld "Chance" verschwendet oder vielleicht doch besser unter​. Kniffel Chance Las Vegas: Huntington Press. Different strategies will also be required should Ladylucks need to achieve a specific target. The goal is to score as many points as possible in all the required categories altogether. Das Spiel ist kommerziell erhältlich, wird jedoch häufig ohne den vorgedruckten Roulette Casino Tips gespielt. It would, however, affect the "Optimal" strategy since scoring a million would have a dramatic effect on the average score. The scoring rule for this category means that a player only scores if at least three of the five dice are the same value.

Kniffel Chance Video

Kniffel - die beste Ausgangslage für eine große Straße Die Kniffel, Yahtzee oder Würfelpoker mit Regeln und Spielablauf. für einen "​billigen Wurf" das Feld "Chance" verschwendet oder vielleicht doch besser unter​. Ich kenne jemanden, der öfter einen Kniffel hatte. In fast jedem Spiel. Ich finde das Spiel frustrierend, haha ich habe da einfach kein Glück. Bei fünf Würfeln gibt es 6 x 6 x 6 x 6 x 6 Möglichkeiten. sechs davon sind ein Kniffel also wieder durch 6. Ja deine Ausführungen sind richtig. Lässt sich eine gewürfelte Kombination in keines der ersten 12 Felder eintragen, so hat der Spieler einmalig durch das Feld "Chance".

Similarly, if a player starts with two "twos" they will be "2 down". 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".

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. The Chance category is often used for a turn that will not score well in any other category.

A Yahtzee occurs when all five dice are the same. 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 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.

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". There are two alternative versions of the Joker rule used. In the simpler, alternative version of the Joker rule 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.

The original game rules released in 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 re-copyrighted in The winner is the player with the highest total. The rules do not specify what happens in the event of a tie.

The probability of completing a Yahtzee depends on whether a particular Yahtzee is required or any Yahtzee. The probability of completing any Yahtzee is shown in the following table.

The strategy is to keep any Yahtzee, four-of-a-kind, three-of-a-kind or pair that is thrown and re-roll the others. With two pairs, either can be chosen.

So if a three-of-a-kind is thrown on the first roll, the chance of completing a Yahtzee increases from 4. This section describes the last round strategy where there is no possibility of a Yahtzee bonus i.

These are the simplest situations to analyse, although even at this late stage the strategy may vary depending on the game situation.

If the final box is in the upper section, the strategy is to keep any of the number required and re-roll the others. On average 2. The distribution is as follows:.

There are slightly different strategies depending on whether a player is simply just trying to get a three-of-a-kind or if they are trying to maximize their average score.

Different strategies will also be required should a specific target be needed to achieve. The strategy to maximize the chance of getting a three-of-a-kind involves keeping any three-of-a-kind that is rolled.

If a three-of-a-kind is rolled then after the first throw the player should keep any other 5s and 6s, while after the second throw the player should keep any other 4s, 5s, and 6s.

An example is with the player keeps and throws the other 2. If a three-of-a-kind is not rolled the player should keep any pair that is rolled and re-roll the other dice, with two pairs the player should keep the higher pair, and with no pair the player should keep the highest die.

Following this strategy gives a 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 three-of-a-kind a guaranteed score of 15 but keeping 66 maximizes the expected average score The situations where the strategy to maximize the average score differs are all after the first throw and are as follows: keep 33, 44, 55, 66 rather than or , , keep 4, 5 or 6 rather than 11, keep 5 or 6 rather than 22, keep rather than For example, with the player should keep 6 rather than 22 and with they should keep not If they follow the strategy to maximize the average score they will get a three-of-a-kind As with three-of-a-kind there are slightly different strategies depending on whether a player is simply trying to get a four-of-a-kind or they are trying to maximize their average score.

Different strategies will also be required should they need to achieve a specific target. The strategy to maximize their chance of getting a four-of-a-kind involves keeping any four-of-a-kind that they have.

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.

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. The game consists of thirteen turns. Each player rolls the dice up to three times in each turn.

Next to the board there is a list of all the required categories. The goal is to score as many points as possible in all the required categories altogether.

All categories in the list must be filled. List of categories and the board with chat and player list windows hidden.

Players are obliged to roll the dice at least once first roll is automatic , but the two remaining rolls are optional.

Subsequent rolls are performed with dice you chose not to put aside, which is done by clicking on them. Only dice left on the table can be rolled.

After completing your turn, it is time to choose which category should be scored. Categories are called simple , e. If you select the wrong category, your turn will be underscored.

For example, if you rolled two two pair , but selected three of a kind , then you are scored 0 points in the three of a kind category.

In some instances you have to choose between two categories. Examples of scoring for figure category. Chance is scored by simply adding the value of all the dice, regardless of which numbers they land on.

Kniffel Chance - Navigationsmenü

Daraus ergibt sich eine mittlere Punktzahl von 9, Ebenfalls ist es eine einfache Rechenaufgabe, ob man für einen "billigen Wurf" das Feld "Chance" verschwendet oder vielleicht doch besser unter "Einser Sammeln" eine "Null" einträgt. Für den 3. Hat jeder Spieler seine 13 Runden absolviert, werden als erstes die erzielten Punkte in Teil 1 addiert. Hat ein Spieler im Feld "Kniffel" bereits etwas eingetragen - und sei es nur eine "Null" - so darf er den 2. Am besten bewertet 1 Mensch ärgere Dann kann statt des Zwillings auch die 6 behalten werden.

Kniffel Chance Video

wie man... IMMER DIE SECHS WÜRFELT (Lifehack) Namespaces Article Talk. The original version of the logo was used throughout the entire period that the game was produced solely by the Lowe company. The following table shows the probability of getting certain scores using the "Optimal" strategy. Preferences In preferences you can set the user name you want to use when saving your Network Scores. You shouldn't wait Kniffel in 2nd column too long, otherwise you may later may miss some other important points in 2nd column. He will Keep all large and small straights. Kategorien : Spiel er Würfelpoker. With two Trading Handel, either can be chosen. Als Wert wird die Booble Shoter aller Würfel eingetragen. Das empfohlene Mindestalter Kniffel Chance 8 Jahren sollte berücksichtigt werden, da das Spiel den Teilnehmern Casino Games App einiges an Denkarbeit und Taktik abverlangt. Als erstes fragen wir uns, welche der 3 Würfel eine 6 zeigen. Bei Kniffel kann aber jedes Kästchen nur einmal mit einem Würfelwert belegt werden, in diesem Beispiel wäre nun das Feld für Fünfen ausgefüllt und Mensa Casino Uni Frankfurt weitere Eintragungen gesperrt. Bei einem Zwilling und drei Einlingen wird nur der Zwilling behalten, wenn es sich dabei um Vieren, Fünfen oder Sechsen handelt. Ansichten Lesen Bearbeiten Quelltext bearbeiten Versionsgeschichte. Geschicktes Taktieren und ein guter Mix aus Risikobereitschaft und bedachtem Spiel werden zum Sieg führen. Durch geschicktes Taktieren muss der Spieler zum Spielende über die zu spielenden 13 Runden eine möglichst Bounty Hunter Authority Punktzahl erzielt haben. Im englischsprachigen Raum ist zudem die Version Erfahrungsbericht Swingerclub bekannt. Ist jedoch ein Feld bereits belegt, so kann der betreffende Eintrag nicht mehr korrigiert werden.

You can see different combinations of dice which score in the left side of the game window. You roll dice at most three times and to roll the dice, press the 'Roll' button.

Prior to each subsequent roll you may choose to hold or release any combination of dice. You hold a die by clicking it with the mouse.

You release a die by clicking it again. When pressing 'Roll', only dice that are not locked will be rolled again.

Latest after the third roll, you must select which score you will mark by pushing score buttons. After scoring you will throw again.

There are 52 different scores and the game is finished when all are marked. Scoring You can score in any of the scoring buttons that are not already selected.

The score is indicated as you move the mouse cursor over the button and those buttons which are not scoring for your throw are disabled.

Once you click a button, your score is registered. First six top rows score according to the number of dice showing the number of 1's, 2's and so on.

For example if you throw throw 4 4 4 4 5 6 you score 16 points if you select it as fours. Bottom rows score if your throw belongs to one of these special cases: Three of a Kind 3 dice of the same kind sum of all the dice Four of a Kind 4 dice of the same kind sum of all the dice FullHouse 2 dice of one kind, 3 of another 25 points Small Straight 4 consecutive numbers 30 points Large Straight 5 consecutive numbers 40 points Kniffel 5 dice of the same kind 50 points Chance any combination sum of all the dice In the top section first six rows you get a bonus 35 points in any of the four columns if your subtotal for that column is greater than or equal to When selecting scores you have to proceed scoring in the first column from top to bottom.

Second column is filled from bottom to top and forth can be filled in any order. Third column can be filled in any order but it will score only after the first throw.

If you can't select any score which will give you points you have to mark zero to that scoring row of the first or second column which is next in turn or to any row of the third or fourth column.

Scores List of ten best scoring games is saved on the game server. In the simpler, alternative version of the Joker rule 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.

The original game rules released in 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 re-copyrighted in The winner is the player with the highest total. The rules do not specify what happens in the event of a tie.

The probability of completing a Yahtzee depends on whether a particular Yahtzee is required or any Yahtzee. The probability of completing any Yahtzee is shown in the following table.

The strategy is to keep any Yahtzee, four-of-a-kind, three-of-a-kind or pair that is thrown and re-roll the others. With two pairs, either can be chosen.

So if a three-of-a-kind is thrown on the first roll, the chance of completing a Yahtzee increases from 4. This section describes the last round strategy where there is no possibility of a Yahtzee bonus i.

These are the simplest situations to analyse, although even at this late stage the strategy may vary depending on the game situation.

If the final box is in the upper section, the strategy is to keep any of the number required and re-roll the others. On average 2.

The distribution is as follows:. There are slightly different strategies depending on whether a player is simply just trying to get a three-of-a-kind or if they are trying to maximize their average score.

Different strategies will also be required should a specific target be needed to achieve. The strategy to maximize the chance of getting a three-of-a-kind involves keeping any three-of-a-kind that is rolled.

If a three-of-a-kind is rolled then after the first throw the player should keep any other 5s and 6s, while after the second throw the player should keep any other 4s, 5s, and 6s.

An example is with the player keeps and throws the other 2. If a three-of-a-kind is not rolled the player should keep any pair that is rolled and re-roll the other dice, with two pairs the player should keep the higher pair, and with no pair the player should keep the highest die.

Following this strategy gives a 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 three-of-a-kind a guaranteed score of 15 but keeping 66 maximizes the expected average score The situations where the strategy to maximize the average score differs are all after the first throw and are as follows: keep 33, 44, 55, 66 rather than or , , keep 4, 5 or 6 rather than 11, keep 5 or 6 rather than 22, keep rather than For example, with the player should keep 6 rather than 22 and with they should keep not If they follow the strategy to maximize the average score they will get a three-of-a-kind As with three-of-a-kind there are slightly different strategies depending on whether a player is simply trying to get a four-of-a-kind or they are trying to maximize their average score.

Different strategies will also be required should they need to achieve a specific target. The strategy to maximize their chance of getting a four-of-a-kind involves keeping any four-of-a-kind that they have.

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.

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.

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