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# An Exploration Of The Theoretical Support Of The 24 Game

991 words - 4 pages

Mathematical Exploration: The 24 game

An exploration of the theoretical support of the 24 game

An introduction to the 24 game:

Overview:
The 24 game is a mathematical card game which originated from China in the 1960s and popularized in China and America later. It is a game which required its players to make fast calculations, and it can be competitive. After years of spreading and development, the game has derived into a lot of different rules. In this research paper, the topic is mainly focused on the original rule.

Rules:
In 24 game, the players are using a standard card deck where the jokers are eliminated from the card deck. By randomly selecting 4 cards from the 52 card deck, the ...view middle of the document...

In a standard card deck with jokers eliminated, there are in total 52 cards. When four cards are randomly picked from this standard deck, to calculate how many combinations of cards there are, the equation shown below can be made:
(52¦4)=52!/4!(52-4)!=270725

Which means, in a card deck, one can get 270725 different ways of combination. However, since the order of the picked four cards does not matter in the game, one thing that needs to be noticed is in this calculation, different suits of the cards (clubs ♣, diamonds ♦, hearts ♥, spades ♠) with the same level are counted as different cards. Which in 24 game, suits do not affect the calculation, therefore, the equation has to be changed in to:
(█(13@4))+(13¦3)×(3¦1)+(13¦2)×((2¦1)+(1¦1))+(13¦1)
=715+286×3+78×3+13
=1820

The equation above gives the correct total number of independent combinations of the 24 game which order and suits does no matter.

In the equation,(█(13@4)) represents that when 4 cards are randomly picked, the 4 picked cards are all in different rank levels; (13¦3)×(3¦1) represents there are 3 different rank levels while 2 cards with the same rank level; (13¦2) ×((2¦1)+(1¦1)) represents there are 2 different rank levels which means there are two pairs of cards with the same rank level; and finally (13¦1) represents that all four cards are in the same rank level.

In 1820 different ways of combination, it is known that many of them have no solution. Although it is impossible for one to calculate each combination and distinguish solvable and unsolvable questions due to the huge amount of work, it is possible to program a computer...

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