# 24 Game

The 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. Addition, subtraction, multiplication, or division, and sometimes other operations, may be used to make four digits from one to nine equal 24. For example, card with the numbers 4,7,8,8, a possible solution is the following: (7-(8/8))*4=24.

The game has been played in Shanghai since the 1960s, using ordinary playing cards.

A Sample 24 Game Card

## Original version

The original version of 24 is played with an ordinary deck of playing cards with all the face cards removed. The aces are taken to have the value 1 and the basic game proceeds by having 4 cards thrown and the first player that can achieve the number 24 exactly using only addition, subtraction, multiplication, division, and parentheses wins the hand. Some groups of players allow exponentiation, or even further operations such as roots or logarithms.

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; players are eliminated when they no longer have any cards.

A slightly different version includes the face cards, Jack, Queen, and King, giving them the values 11, 12, and 13, respectively.

## Strategy

Mental arithmetic and fast thinking are necessary skills for competitive play. Pencil and paper will slow down a player, and are generally not allowed during play anyway.

There are $\tbinom {4+13-1}4=1820$ four-card "hands" when played with a standard 52-card deck. It is impossible to make 24 with some of these, for example 1,1,1,1 while others can be difficult, such as 3,3,8,8: (8/(3-(8/3))=24).