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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
so that the end result is 24. For example, for the numbers 4, 7, 8, 8, a possible solution is
. Note that all four numbers must be used exactly once.
The problem has been played as a
card game
A card game is any game that uses playing cards as the primary device with which the game is played, whether the cards are of a traditional design or specifically created for the game (proprietary). Countless card games exist, including famil ...
in
Shanghai
Shanghai, Shanghainese: , Standard Chinese pronunciation: is a direct-administered municipality and the most populous urban area in China. The city is located on the Chinese shoreline on the southern estuary of the Yangtze River, with the ...
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 f ...
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. In a standard 52-card pack of the English pattern, ...
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 and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
,
subtraction
Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
,
multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
,
division, 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. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...
) wins the hand. Some advanced players allow
exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
,
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
,
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
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 armed conflict between the armed forces of states, or between governmental forces and armed groups that are organized under a certain command structure and have the capacity to sustain military operations, or between such organi ...
. 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
People and fictional characters
* Jack (given name), a male given name, incl ...
,
Queen
Queen most commonly refers to:
* Queen regnant, a female monarch of a kingdom
* Queen consort, the wife of a reigning king
* Queen (band), a British rock band
Queen or QUEEN may also refer to:
Monarchy
* Queen dowager, the widow of a king
* Q ...
, and
King
King is a royal title given to a male monarch. A king is an Absolute monarchy, absolute monarch if he holds unrestricted Government, governmental power or exercises full sovereignty over a nation. Conversely, he is a Constitutional monarchy, ...
, giving them the values 11, 12, and 13, respectively.
In a variation of the game played with a standard 52-card deck, there are
four-card combinations.
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
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...
, it is possible to get 24 as
.
See also
*
Krypto (game)
Krypto is a card game designed by Daniel Yovich in 1963 and published by Parker Brothers and MPH Games Co. It is a mathematical game that promotes proficiency with basic arithmetic operations. More detailed analysis of the game can raise more com ...
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