The 24 puzzle is an arithmetical puzzle in which the objective is to find a way to manipulate four

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

so that the end result is 24. For example, for the numbers 4, 7, 8, 8, a possible solution is

(7-(8\div8))\times4=24

. The problem has been played as a

card game A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games (such as poker). A small number of card g ...

Shanghai Shanghai (; , , Standard Mandarin pronunciation: ) is one of the four direct-administered municipalities of the People's Republic of China (PRC). The city is located on the southern estuary of the Yangtze River, with the Huangpu River flowin ...

since the 1960s, using

playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a ...

s. It has been known by other names, including Maths24. A proprietary version of the game has been created which extends the concept of the basic game to more complex mathematical operations.

Original version

The original version of 24 is played with an ordinary deck of playing cards with all the

face card In a deck of playing cards, the term face card (US) or court card (British and US), and sometimes Royalty, is generally used to describe a card that depicts a person as opposed to the pip cards. They are also known as picture cards, or until the ...

s removed. The aces are taken to have the value 1 and the basic game proceeds by having 4 cards dealt and the first player that can achieve the number 24 exactly using only allowed operations (

addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...

subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...

division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military * Division (military), a formation typically consisting ...

, and

parentheses A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...

) wins the hand. Some advanced players allow

exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

, roots,

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

s, and other operations. For short games of 24, once a hand is won, the cards go to the player that won. If everyone gives up, the cards are shuffled back into the deck. The game ends when the deck is exhausted, and the player with the most cards wins. Longer games of 24 proceed by first dealing the cards out to the players, each of whom contributes to each set of cards exposed. A player who solves a set takes its cards and replenishes their pile, after the fashion of

War War is an intense armed conflict between states, governments, societies, or paramilitary groups such as mercenaries, insurgents, and militias. It is generally characterized by extreme violence, destruction, and mortality, using regular o ...

. Players are eliminated when they no longer have any cards. A slightly different version includes the face cards,

Jack Jack may refer to: Places * Jack, Alabama, US, an unincorporated community * Jack, Missouri, US, an unincorporated community * Jack County, Texas, a county in Texas, USA People and fictional characters * Jack (given name), a male given name, ...

Queen Queen or QUEEN may refer to: Monarchy * Queen regnant, a female monarch of a Kingdom ** List of queens regnant * Queen consort, the wife of a reigning king * Queen dowager, the widow of a king * Queen mother, a queen dowager who is the mother ...

, and

King King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king. *In the context of prehistory, antiquity and contemporary indigenous peoples, the ...

, giving them the values 11, 12, and 13, respectively. In the original version of the game played with a standard 52-card deck, there are

\tbinom 4=1820

four-card combinations.https://www.4nums.com/24_game/facts/

Expansion to more complex operations

Additional operations, such as square root and factorial, allow more possible solutions to the game. For instance, a set of 1,1,1,1 would be impossible to solve with only the five basic operations. However, with the use of factorials, it is possible to get 24 as

(1+1+1+1)!=24

References

External links

24game.github.io: an online, open-source version of the 24 puzzle

List of all possible solutions to the puzzle
{{Children's card games Card games introduced in the 1960s Mathematical games Brain training video games Year of introduction missing Card games for children

Original version

Expansion to more complex operations

See also

References

External links