Notes. Alan Zhong

24 Game

I played 24 for the first time last weekend. The game is pretty simple, you’re given 4 cards from a deck, 1 through 13, and must make 24 using a combination of basic operations or additional operations. For example if we drew 5, 10, Q, K, that would translate to 5, 10, 12, 13 which we could make 24 doing (10+5-13)×12.

Computationally there are two straight forward ways of solving this.

  1. Iterate through all bracket and operation placements until you find the solution
  2. Solving for 4 numbers is the same as solving for 3 given every pair in the original 4.

However these strategies are ineffective in real play because not only do you have to solve for 24, you have to be the fastest. The median solution time for all solvable draws is 9 seconds.

cdf-average-time.png

In general, draws with more solutions tend to be easier and solved faster.

solved_rate_vs_amt.png

Looking at a subset of skilled players, the median solution time for all solvable draws drops to 3 seconds.

skilledcdff.png

Fortunately for mere mortals, draws with more solutions tend to be easier and solved faster even for skilled players.

solved rate vs skilled players.png

At a median of 3 seconds per draw, a skilled player would beat another player trying to enter combinations of numbers into a computer. Anecdotally, I’ve seen players call out the solution within a split second of the cards being turned over, sometimes by the dealer themselves. One possible strategy is to watch for tendencies in players to be better at specific operations, choose to give up those, and focus on other operations. Even skilled players have difficulty with harder operations:

solvetime_vs_complexity.png