Small-Grid Wraparound Cellular Automata

The problem asks how long a small, horizontally and vertically wrapping cellular-automaton grid can pass through distinct patterns before repeating. Each cell is alive or dead, and its next state depends only on how many of its eight neighbors are alive - not on its own current state. Because a finite grid has finitely many configurations and updates deterministically, every sequence eventually loops. The aim is to find the greatest possible number of unique states for each grid size, over all starting patterns and rules. On a 3×3 grid, every cell is a neighbor of every other cell, so the possible changes are unusually restricted; the maximum is just four distinct states.

For larger grids, the analysis uses computation. Equivalent starting patterns can be eliminated by accounting for translations, reflections, and rotations; inversion can also pair a starting pattern with a complementary rule. These reductions shrink the 4×4 search to 206,080 pattern-rule combinations, yielding a maximum of 14 unique states. The reported maxima rise sharply for 5×5, to 916, but only to 1,717 for 6×6 - far less growth than expected. The author describes using brute-force enumeration for 4×4 and later computational optimization, with AI-assisted coding, for the larger cases. The unexpected 6×6 result prompts speculation that the grid’s divisibility may affect the behavior, though no explanation is established. The account combines a precise mathematical question and its computational answers with the personal story of how the 4×4 solution helped launch the author’s programming career.