Analysing 2048 on a 3×3 board

In 2018, the author wrote an analysis of the 3x3 version of the popular puzzle game 2048. The author aimed to determine the optimal target tile value that a perfect player could achieve on a smaller 3x3 grid. The game rules are simple, and the state space of the 3x3 board is manageable, containing only (k+1)^9 possible game states, where k is the desired target tile value.

This allowed for a comprehensive analysis of the game. The author emphasized that 2048 is a monotonic game, meaning that the sum of all tiles on the board strictly increases with each move, eliminating the possibility of infinitely looping games. The goal was to calculate the probability of a perfect player successfully making a specific target tile, such as 2048.

To achieve this, the author assigned a value to each board position, representing the probability of successfully making the target tile from that position. Two probabilities, pvalue and cvalue, were considered, with pvalue representing the chance of the player making the target tile and cvalue representing the chance of the computer making the target tile.

The author chose to use a recursive dynamic programming approach to solve the problem, as it efficiently calculated the values for each board position without the risk of stack overflow, which could occur in a non-recursive approach.