# Card Game: 24

1. Dec 12, 2007

### davee123

The Game
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We used to play a card game called 24 that a friend of mine invented. Remove all the face cards from the deck so that you're left with nothing but 2-10's and Ace's. Ace's are worth 1, everything else is at its face value. Each of the two players (this only really works well with 2 players) takes half the deck. Then, you start rounds wherein each player lays out 2 cards from their stack, such that there are 4 cards total on the table. Now, as quickly as you can, you attempt to mathematically make the number 24 using *only* the numbers on the table. Whoever comes up with the formula for 24 first takes all 4 cards and replaces them randomly into their stack (or at the bottom, whatever). Play continues until one person collects the entire deck and is declared the winner!

While playing, we found that certain functions ought to be disallowed because they used implied numbers. For instance, the square root function is disallowed because there's an implied "2" in there. The "sine" function (if anyone was ever crazy enough to use it) would be disallowed because it implied a "pi" and a "360" somewhere in there. Factorials are disallowed because they imply *every* positive integer less than their operand. Etc. Hence, the allowed functions always took 2 numeric operands, and are:

- Subtraction
- Multiplication
- Division
- Power
- Root (with explicit base)
- Logarithm (with explicit base)
- Modulus

Also, each number must be used exactly once. So every solution must include exactly 3 operators. For example, you might have:

2, 6, 7, 9 ==> ((9 - 7) + 2)*6 = ((2) + 2) * 6 = 24
1, 2, 3, 8 ==> (Log base 2 of 8) * (8 / 1) = (3) * (8) = 24
3, 3, 4, 8 ==> (root 3 of 8) * (3 * 4) = (2) * (12) = 24
1, 4, 5, 10 ==> 5 ^ (10 % 4) - 1 = 5 ^ (2) - 1 = 25 - 1 = 24

The Question(s)
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So, this got me thinking. We chose the number 24 as a "goal" because it's got lots of divisors, and is "within reach" (IE, it's sometimes possible to simply add up all the numbers available and get 24). But even so, there are some combinations were 24 isn't attainable. For example, if the numbers were 3, 5, 7, and 7, you can't get 24 out of them, no matter how hard you try. In fact, of the 715 different possible hands that you could have, 94 of them are impossible. So what if we set a different "goal" number?

1) (easy) With a goal number of 473, there are 5 different sets of cards that will allow you to come up with an answer:

A) 2,3,6,8
B) 2,3,8,9
C) 3,4,4,6
D) 3,4,4,9
E) 6,7,8,10

For each of the above sets of cards, find a method for obtaining 473 using the 4 numbers.

2) (hard) What is the lowest goal number such that it is unattainable, no matter what 4 cards you draw?

3) (very hard) What is the lowest goal number such that there is only ONE set of cards that will allow you to reach the goal?

4) (more a matter of trivia) What goal number is exactly equivalent to the number of sets of cards that allow you to reach that goal number?

DaveE

Last edited: Dec 12, 2007
2. Jan 16, 2009

### davee123

I just realized I never answered this question!

1)
A) 3^6 - 2^8 = 473
B) 9^3 - 2^8 = 473
C) 3^6 - 4^4 = 473
D) 9^3 - 4^4 = 473
E) 6*8*10 - 7 = 473

2) 461, unreachable no matter what cards you're dealt.

3) 419, only attainable with 1, 6, 7, 10 => 6*7*10 - 1

4) 225 has 225 combinations of cards that can get to 225.

3. Jan 16, 2009

### Jimmy Snyder

Makes sense to me. Fights could break out if you had 3 players all vying for half the deck. Are roots allowed or not? Your explanation says both.

4. Jan 16, 2009

### davee123

Effectively, you're allowed to use a root as if it takes two parameters, NOT as though it takes 1 parameter. Each mathematical function that's allowed in the game MUST take two parameters. In the case of "root", it takes two parameters, the number that you're rooting (not sure of the name offhand), and the base. But if you wanted to use "square root" as though it took only ONE parameter (the number being rooted), then that's disallowed. For example, let's say you had: 9, 2, 2, 4.

Valid use of root:
SQRT(9) * 2*4 = 24

Invalid use of "square root":
SQRT(9) * (2+2+4) = 24

So any time you want to use a root function, you have to explicitly specify the base using one of the numbers you've been given.

DaveE

5. Jan 16, 2009

This reminds me of a little arithmetic minigame called "Rydia's Mathemagic Minute" from Final Fantasy, only with more possibilities. I think it's pretty fun. You said it works well with 2 players, wouldn't it be funner with 4 players?.. although tough luck trying to find 4 people willing to play a math card game :)

6. Jan 16, 2009

### davee123

I think the problem we had with adding more players was that it just got a little wonky. 3 players would mean either that you'd make 24 out of *6* cards rather than 4, or that players would alternate who put in 1 or 2 cards. And when 1 player was out, things would revert back to the normal 2-player rules. It also made for strange possibilities in how many cards you could have in your stack. It was possible with odd number of players that you'd have an odd number in your stack, and couldn't contribute the necessary number of cards from your pile when needed. And since a 4-player game would probably lead to having a 3-player game, that was similarly discouraging.

These aren't insurmountable problems, to be sure-- but as stated, it works best with 2 people.

And there's the real reason we never played a 3+ player game! To date, I've only ever managed to sucker 4 people into playing this game with me. My friend came up with it in high school back in the early 90's I guess, and we'd play it upon occasion while waiting for class or something similar. Anyway, I guess I don't find too many people who are interested anymore...

It never occurred to me to look online for this game, but it appears that a similar game (possibly the one to inspire my friend to come up with his version) was made in 1988 using custom-made cards, each with 4 numbers already printed on them:

http://en.wikipedia.org/wiki/24_Game

Of course it didn't allow funky functions like modulus, logarithms, etc. And it probably was always possible to reach 24 with the given card, unlike our variant that used effectively random numbers between 1 and 10.
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DaveE

Last edited: Jan 16, 2009