# 24 Game

The 24 Game is an arithmetical card game in which the object 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. Robert Sun commercialised the game and copyrighted it in 1988, introducing dedicated game cards bearing four numbers each, and sold it through his company, Easton, Pennsylvania-based Suntex International Inc. There are nine official editions of Suntex's 24 Game. A tournament-style competition, referred to as the 24 Challenge, is based on the 24 game.

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.

## Overview of the commercial game

Cards are divided into three levels of difficulty. One-dot cards (with a single white dot in each corner, indicating an easy problem) are often solved by simple addition, or contain three digits that can make 24, plus a 1 (in which case any other digit could be multiplied or divided by 1 to create the same digit). Two-dot cards (with two red dots) are slightly more difficult, and often require more multiplication and division than one-dot cards. Three-dot cards (with three yellow dots) are the most difficult cards, often having only one solution. In most decks of Math 24 cards, the ratio of one-dot cards to two-dot cards to three-dot cards is 1:2:1.

### Variations

There have been many variations on the original Single Digits Edition 24 game, including standard 96-card editions and 48-card travel-sized editions. Some variations include:

• Double Digits Edition — Cards include 2-digit numbers, from 1 to 24 (excluding 19), leading to more difficulty as not as many students are as familiar with multiples of larger numbers.
• Variables — Cards have two wheels, each has three numbers with one number "missing." The object is to find a number (any integer 1 - 9) which, when used with the other numbers on each wheel, can make 24 on both wheels.[1]
• Fractions/Decimals, Algebra/Exponents and Integers(negative numbers) versions are also available and are recommended for ages 9 and up.
• Editions for ages 7/8 and up include: Add/Subtract Primer; Multiply/Divide Primer and Factors/Multiples. Primer editions have a "self-check" feature (answer on reverse side)

## 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).

## Tournament

In the 1980s through 2008, there were 24 Challenge tournaments held throughout the United States and in other countries. The tournaments were very popular in Pennsylvania, where many participating schools held school competitions. Each school then sent one or two students to compete at a regional level, and the top four went on to the state level Championship. In many states, there was no state-level competition.

Official, Suntex-sponsored tournament events were suspended after the 2008 season due to lack of corporate funding, which had provided 24 Challenge Math Program materials to participating schools. Today, many schools continue to conduct their own 24 Challenge events. Over a period of months preceding the event, teachers use the 24 Game in their classrooms to help students hone their mathematics skills. Schools or districts that hold regional or district-wide events schedule playoffs to determine which students will advance to their Championship event.

## Platinum Series

Several variations of Math 24 cards exist, and are used in the Platinum stages of tournaments. This level of play is highest in the 24 game, and only 7/8 graders can participate. The cards used are Algebra/exponents, fractions/decimals, and integers. They can be purchased at 24 game.com ( link located here [1]). These numbers are treated the same way as regular digits, and must be used in a solution once.

A special "integers" deck uses negative digits alongside positive numerals. (Cards in this version may be solved for positive or negative 24.) However, in the tournament, positive 24 must be found.

Algebra version cards contain values with a variable, such as 3y or 2x-4. In solving a problem, the player must state what each variable represents, then give the solution using that variable in it. Cards may contain more than one variable on a side; three-dot cards commonly use x, y, and z all on one card.

The "Exponent" version of Math 24 integrates roots and powers into game play. These cards have a special center marking, indicating that one digit (or result from a previous equation) must be squared, cubed, or have the square root or cube root taken. This results in a card requiring four operations, instead of the usual three. For example, a card with the digits 2, 3, 4, and 8 might be solved by stating that 2x8=16, the square root of 16 is 4, 4+4=8, and 8x3=24.

The last form of platinum play are the "fraction" cards. They involve fractions along with usually whole numbers, except in the case of some level three cards.