Evolomino is NP-complete

TL;DR

The paper proves Evolomino is $NP$-complete via a polynomial-time reduction from $3$-SAT, using gadgets (variable/wire, negation, split, clause, crossover) to encode SAT instances on Evolomino boards. The construction yields a board of size $O(m^2+n) × O(m^2+n)$ and is parsimonious, so Counting Evolomino is $#P$-complete as well. The authors also show Evolomino is in $NP$ by polynomial-time verification and discuss implications and potential future directions for algorithmic approaches and restricted-case studies. These results place Evolomino among hard pencil-and-paper puzzles and provide a formal framework for analyzing similar combinatorial puzzles.

Abstract

Evolomino is a pencil-and-paper logic puzzle popularized by the Japanese publisher Nikoli (like Sudoku, Kakuro, Slitherlink, Masyu, and Fillomino). The puzzle's name reflects its core mechanic: the shapes of polyomino-like blocks that players must draw gradually "evolve" in the directions indicated by pre-drawn arrows. We prove, by reduction from 3-SAT, that the question of whether there exists at least one solution to an Evolomino puzzle satisfying the rules is NP-complete. Since our reduction is parsimonious, i.e., it preserves the number of distinct solutions, we also prove that counting the number of solutions to an Evolomino puzzle is #P-complete.

Figures (10)

• Figure 1: Example of an Evolomino puzzle
• Figure 2: Variable and wire gadget
• Figure 3: Negation gadget
• Figure 4: Split gadget
• Figure 5: Clause gadget

Theorems & Definitions (5)

• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary