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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:

- Addition

- 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

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# Card Game: 24

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