24 Game

This article is about the mathematical game. For the game based on FOX's television series, see 24: The Game.

The 24 Game is a mathematical 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 an 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^[1], 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, 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, 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.^[2]
• 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

Mastering this game requires fast thinking, but with enough practice, the ability to spot useful patterns increases. Some experienced players can make 24 with almost any four numbers. There are several sequences that lead to 24 that players frequently try to obtain. For example, the solutions 6 x 4, 8 x 3, 2 x 12, 18 + 6, 16 + 8, and 10 + 14 often appear.

There are many common patterns that help a player to acquire a solution faster. It is beneficial to look for common multiplication patterns, such as 12 x 2, 8 x 3, and 6 x 4 in a card to equal 24. Similarly, multiplying a digit by another digit that is one more or less than the usual multiplicitive pair (for example, 4 x 5 or 4 x 7 instead of 4 x 6) then adding or subtracting the first number (20 + 4 or 28 - 4) is a common strategy for reaching 24. This strategy comes from the law of distributivity from elementary algebra. These are some more examples:

6 x 4 = ...

     = 6 x (3 + 1) = 6 x 3 + 6
     = 6 x (5 - 1) = 6 x 5 - 6
     = (5 + 1) x 4 = 5 x 4 + 4
     = (7 - 1) x 4 = 7 x 4 - 4

8 x 3 = ...

     = 8 x (2 + 1) = 8 x 2 + 8
     = 8 x (4 - 1) = 8 x 4 - 8
     = (7 + 1) x 3 = 7 x 3 + 3
     = (9 - 1) x 3 = 9 x 3 - 3

Possible solutions can be found on the back of 24 game deck boxes.

The number 1 is a powerful digit on a card. As stated above, a 1 may be used at any time to multiply or divide any number to equal itself. This is important in cards where only three digits are needed to create the value of 24. A player may simply multiply (or divide) 24 by 1 to create the final result of 24. If a 1 is not readily provided on the card, two numbers may be subtracted (such as 8-7) or divided (6/6) to make a 1.

Pencil and paper are generally not allowed during play; using such would only slow down a player anyway. Mental math is a necessary skill for playing this game. One good strategy is taking advantage of odd numbers. Sometimes when I play when there is two or four odd numbers I try to make them even. With just one or three odd numbers I try to make them a three so I can multiply by eight.

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-sposored 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.

In tournament play, several extra rules apply:

• A player, upon tapping the card, must state the final operation they used to reach 24 first. They then proceed to list the operations in normal order, subsequently repeating the final operation. If a player forgets to state the final operation first, they receive a penalty flag. If, in mentioning all steps they used to reach the card, they do not use the same final operation, they receive a penalty flag.
• Tapping a card with one finger, four or five fingers, or slapping with the whole hand results in a penalty flag, or a relinquishment of a card.
• Players must keep their hands (usually, only the tapping two or three fingers) on the blue region in the mat, about eight inches away from the card. Touching inside this area results in a penalty flag.
• Fake cards with no solution were included in "Tournament Decks"* in the first round to discourage impulsive tappers. Tapping one of these fake cards results in an automatic penalty flag. *(Special Tournament Decks (with non-working cards) were only used at official Suntex-sponsored events, and were not distributed to the general public or sold.)

When a player receives three penalty flags, they are disqualified from further play in that round, but they keep any points they have earned up to that point.

In order to win, the player must earn more points than everyone else in the competition. Usually there are two rounds. The players move from table to table playing other students in their same level. For each card that they receive, they get more points. The points are distributed based on the number of dots found in the corner of each card. The player at the end with the most points wins.

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.

Online Game

In addition to the game produced by Suntex on small cards, there is also a 24® game online at firstinmath.com. The First In Math® Online Program (FIM) was introduced in 2002. The backbone of the FIM program is the 24® Game, but the 24 editions available online range from Addition Only to multi-step Algebra, and are organized into eight SKILL SETS™. Each Skill Set comprises three sequential 24 Games, labeled 1, 2 and 3. Each game builds upon the skills acquired in previous games. After completing certain Skill Set requirements, students can play BONUS GAMES. Bonus games offer various types of skill-building activities, such as sequencing and working with tens. Bonus Games are targeted to skills students need to master at each grade level, based on common core math standards.

A Pre-test/Post-test module called JUST THE FACTS records each student’s proficiency with basic facts, as players try to complete the addition, subtraction, multiplication, and division tables in under 5 minutes. Three KNOW & SHOW modules provide a unique way for students to experience the type of problems found in standardized tests. Students solve problems from nine categories that tie to national math standards. A workout GYM module helps build automaticity with addition, subtraction, multiplication and division facts, and progress to include fractions, decimals and integers. MEASUREMENT WORLD showcases games and activities that focus on Money, Time, Length, Distance, Area, Volume and Weights.

The First In Math® site allows players to compete individually or as a classroom "Team" and then compare their score to other players on their Team, in their school, school district, state and nationally. Players earn virtual "stickers" that accrue to their personal and classroom score by correctly solving problems on the site.

With the increasing popularity of smart phones, a number of 24 applications have been ported to mobile devices such as S60 phones with J2ME support, iPhone and android. These games use different user interfaces due to limited screen size, yet they're able to achieve the core objective of Game 24. The officially-licensed version of the 24® Game became available on iTunes in December of 2010. Currently, only the Single Digits version is available.

References

1. ^ Rules of Card Games: Twenty-Four: introduction
2. ^ 24game.com - 24 Game 96 Card Decks

External links

• www.24game.com The Official 24 Game website
• 24 at Pagat
• A fast 24 game solver: both executable program and source code
• Online 24 Game
• 24 Math Practice - A comprehensive iPhone App to practice finding 24
• iMath24s - A 24 game for S60 smartphones
• Slide 24 - The 24 game for iPod Touch and iPhone
• 24.roundskylab.com the iphone version of 24
• Cards Math by virtuesoft.com the iphone version of 24
• TwentyFour! - A multiplayer version of 24 for the iPhone
• 24 game scorers in over 40 different programming languages on the Rosetta Code site
• 24 game solvers in over 23 different programming languages on the Rosetta Code site
• itestu.com/24 An online 24 card game where users are ranked by how fast they solve a set of problems.
• game24.apphb.com An online 24 game and solver.

Also available as a Google Chrome App.