An Integer Linear Programming Model for the Evolomino Puzzle

TL;DR

The rules of Evolomino are formalized as an integer linear programming (ILP) model, encoding block evolution, connectivity, and consistency requirements through linear constraints through linear constraints.

Abstract

Evolomino is a pencil-and-paper logic puzzle published by the Japanese company Nikoli, renowned for culture-independent puzzles such as Sudoku, Kakuro, and Slitherlink. Its name reflects the core mechanic: the polyomino-like blocks drawn by the player must gradually "evolve" according to the directions indicated by arrows pre-printed on a rectangular grid. In this paper, we formalize the rules of Evolomino as an integer linear programming (ILP) model, encoding block evolution, connectivity, and consistency requirements through linear constraints. Furthermore, we introduce an algorithm for generating random Evolomino instances, utilizing this ILP framework to ensure solution uniqueness. Computational experiments on a custom benchmark dataset demonstrate that a state-of-the-art CP-SAT solver successfully handles puzzle instances of up to $11 \times 11$ within one second and up to $18 \times 18$ within one minute.

Figures (5)

• Figure 1: Example of an Evolomino puzzle
• Figure 3: An example of a flow used to verify polyomino connectivity
• Figure 4: Example of a translation with $t^{a2}_{-7}=1$, i.e., a 7‑cell backward shift; the copy of the first block inside the second along the arrow is highlighted in blue
• Figure 5: Performance profile of the solvers on a $10 \times 10$ grid (180-second time limit)
• Figure 6: CP-SAT performance: median (line), IQR (shaded), and outliers (dots)