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

Let ni 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:
n1 + n2 + ... + n13 = 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  4 Numbers! or to get 4 Numbers puzzle of the day, everyday.