So I’ll spend a little bit more time on the 24 puzzle I talked about the other day. To describe the puzzle that was posed to me, here’s the game:

- Take a normal card deck, sans jokers
- Assign the values 1-13 for the cards (A = 1, J = 11, Q = 12, K = 13, face otherwise).
- Draw four cards. Arrange in any order necessary, and using the basic operators (+, -,
**, /), form an expression that produces the number 24. For example, {1,2,3,4} can be solved as 1**2*3*4. {4,4,10,10} can be solved as ((10*10)-4)/4.

So I did some dirty figuring, and for the game as described, there are 28561 possible puzzles (duplicates included). Of those 28561, a quick program tells me that 19951 are insoluble. Some of them are obvious, such as {1,1,1,1}, some are a lot harder. As a result, one can only hope to solve the problem 30% of the time. That sucks. If we constrain the puzzle to just digits 1-9 (as suggested by the variation of the game war mentioned in my first mention), we have 6561 possible puzzles. 5% better, but not much.

I’m sort of bummed by this, because I figured this was a game where in most cases there is a solution … in that if I knew with 90% confidence there was a solution for a given set of cards, it would be fun to play to practice doing math in my head … but if I have only a 30-35% chance of actually seeing a problem that can be solved, the challenge then becomes whether or not I am clever enough to figure out if a problem has a solution.

Of course, this is in terms of card combinations, not in terms of actual probability — once you draw a 1, you have a better chance of not drawing a 1 … so the actual analysis is somewhat different. I tried 50000 pseudo-random draws from the deck, and observed the trends in how likely a given hand is to be solved, based on the 13-valued game. The results dropped by a tenth of a percent. Still not of much use. Oh well.

Oh well.