NP-completeness of fill-a-Pix and σ₂ ^p-completeness of its fewest clues problem

Fill-a-Pix is a pencil-and-paper puzzle, which is popular worldwide since announced by Conceptis in 2003. It provides a rectangular grid of squares that must be filled in to create a picture. Precisely, we are given a rectangular grid of squares some of which has an integer from 0 to 9 in it, and our task is to paint some squares black so that every square with an integer has the same number of painted squares around it including the square itself. Despite its popularity, computational complexity of Fill-a-Pix has not been known. We in this paper show that the puzzle is NP-complete, ASP-complete, and #P-complete via a parsimonious reduction from the Boolean satisfiability problem. We also consider the fewest clues problem of Fill-a-Pix, where the fewest clues problem is recently introduced by Demaine et al. for analyzing computational complexity of designing “good” puzzles. We show that the fewest clues problem of Fill-a-Pix is Σ₂ ^P -complete.