# Twenty-Four Adding games

# Twenty-Four

*This page is based on information from Linda Wang, Zachary Dexter, Anthony Lloyd and Alex Fink.*

## Introduction

This Chinese game requires skill at mental arithmetic. It has been played in Shanghai at least since the 1960's and may have originated there, and it is also known in some other cities, for example in Qingdao and Guangzhou. Since the end of the 20th century it has also spread to some places in North America. The basic game for two players will be described first, followed by variations for a larger number of players and extra options.

## Players, Cards and Deal

From a standard international pack remove all picture cards and jokers leaving just 40 cards: the numbers from ace (1) to 10 in each suit.

Shuffle the cards equally between the two players, each having a face down stack of 20 cards which they must not look at until they are played.

## The Play

The two players simultaneously take the top two cards of their stacks and place them face up in front of them, to form a square of four face up cards. Now each player tries to think of a way to combine the four numbers using only addition, subtraction, multiplication and division to get a result of 24. For example if the four cards were 2, 3, 6, 9 then possible solutions would be (9+6-3)×2 or (9-3)×(6-2) or (9×3)-(6÷2).

The first player to think of a solution slaps the table and calls out the solution. If it is correct, the four cards are given to the other player to place at the bottom of his or her stack. Then each player again places two cards face up and players again try to make 24 from the four numbers showing. This process is repeated until one player has no cards. This player is the winner.

If a player slaps the table and then calls out a solution that is wrong, or cannot immediately give a solution, then this player must take the four cards as a penalty.

If neither player can make 24, they may agree after a little while that there is no solution. In this case, each player chooses one of their cards to take back, adds this card to the bottom of their stack, and replaces it with a new face up card from the top of their stack.

Players may shuffle or cut their card stacks any time they wish, but must not look at the cards in their stacks.

In the standard game the four numbers must be combined using *only* the arithmetic operations +, -, ×, ÷, with brackets (, ) if necessary to define the order in which operations are carried out. All four numbers must be used.

## Four Player Game

Four players can play 24 as follows. The deck is divided equally between the players. Each player places a card face up. A player who has a solution slaps the table but says nothing. When three players have slapped the table, the fourth player must select one of them. If the selected player gives a correct solution, the fourth player must take the four cards; if not the selected player must take the four cards. In this version it is possible to bluff by slapping the table when you do not in fact have a solution, in the hope that you will not be selected. As in the two-player game, if no one slaps the table, the players can agree that there is no solution. Then all players put their cards on the bottom of their decks and turn up a new card.

## Card collection version, for two, three or four players

In this version, the first player to call a correct solution takes the four cards and places them under his or her stack. Players who run out of cards can still call solutions and thereby win some cards back. If there are four players with cards, each places one card face up. When there are three players with cards, the player who won the previous set places two cards and the others one each. (To start a three-player game 13 cards are dealt to each player and one card is placed face up; the three players each contribute one card to the first set.) When there are two players with cards each player places two cards. When only one player has cards, this player has won the game.

## Notes

Players should try to show their two cards at the same time as each other, and should place the cards so that they are clearly visible, being careful not to cover the cards with their hands.

The total of 24 is often achieved by grouping the cards to make a two numbers that can finally be multiplied: 4×6, 3×8 or 12×2, but this is not always the case. A surprisingly difficult problem is to make 24 with the numbers 3, 3, 8, 8. Given the limited resources available it may seem impossible at first - most people take a considerable time to find the solution.

## Other Variations

A version of 24 can be played by more than two players. In this case the 40 cards are kept in a single shuffled stack and one player (the dealer) drops four cards face up on the table. In China, it is quite usual simply to play a series of throws in which the first player to find a solution calls it out, or the players agree that there is no solution, without keeping any kind of tally of the overall score. If you prefer to keep score one possible method is as follows. The first player who finds a way to make 24 from the four numbers scores a point. If no solution can be found, the first person to call "no solution" wins a point - but if another player finds a solution after "no solution" was declared, then the player who claimed no solution gets nothing and the finder of the solution scores 2 points.

Some players allow additional operations, such as powers and roots. For example you could use two 3's to get 27 (3 to the power 3). A square root requires a 2 - for example 9 and 2 can then be combined to make 3 (square or 2nd root of 9). Some people even allow logarithms - for example the base 2 log of 8×4 is 5. Factorials are never allowed - each operation must use at least two numbers - and approximations are never allowed. the result always has to be exact, without rounding.

Some players include the Jack, Queen and King in the pack, valued at 11, 12 and 13 respectively, or at 10 each. Some add the black and red Jokers with values of 14 and 15 respectively.

## Other 24 web pages

On the 4 Numbers site you can play 24 against the clock, compare your speed with that of other players, find how many solutions exist for any set of four numbers, and read about the theory of the game and what makes some sets harder to solve than others.

In 1988 Robert Sun, who was brought up in Shanghai, created a version of 24 using special cards each of which shows four numbers, the cards being rated according to difficulty. Linda Wang's father learned the game with standard cards as a child in the 1960's, and says that it had already been known in Shanghai for a while before that. The BoardGameGeek site has pages both on the original game and on Robert Sun's version with special cards.

Wikipedia also has a page on the original game under the name 24 Puzzle.