## How many different situations are there? Let's agree that {1,2,3,4} and {4,3,2,1} are the same

Let n_{i} be the number of cards with value i on it. The number of 24-game situations is the number of non-negative integer solutions to the following equation:

n_{1} + n_{2} + ... + n_{13} = 4

And the answer is = 1820. See explanations here.
Not all quadruples are equally likely to be drawn from a deck of cards.

## How many of the 1820 situations are solvable?

1362 or 74.84% of them are solvable. To get this you have to investigate every situation, there is no short cut. Be very careful that it does not mean that randomly draw 4 cards from a deck of poker, there is a 74.84% chance that it is solvable!
Then what is the chance it is solvable if one randomly draw 4 cards from a deck of poker cards?

To answer this question, we have to answer the following question: what's the chance that you draw {1,2,3,4}, or {6,6,6,6} from a deck of poker cards?

We observe the following facts, let a, b, c, d are *any * 4 different values in {1,2,3,....13}. And let * P()* be the probability drawn from a deck of card, then:

Now we go through all the solvables, add the individual probability of each solvable and conclude that:

The chance that it is solvable is 0.8046, if one randomly draw 4 cards from a deck

## Why 24? Why we don't play 12 the math game or 29 the math game?

24 is the smallest natural number with at least 8 divisors have something to do with it. This means that randomly draw 4 cards
from the deck, there is a decent chance that it's solvable.

A day has 24 hours and a lot of other things might also have something to do with our fancination with 24.

** The following chart shows the number of solvable quadruples and the chance that a randomly drawn quadruple is solvable for n, the math game, n = 1,2,... 100 **

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