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@TECHREPORT{Friedman_spiralgalaxies,
    author = {Erich Friedman},
    title = {Spiral Galaxies Puzzles are NP-complete},
    institution = {},
    year = {}
}

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Abstract:

Introduction Spiral Galaxies puzzles are pencil and paper puzzles which originated in Japan. Each puzzle consists of a grid of squares, and a collection of circles which are the centers of rotationally symmetric polyominoes which tile the grid. The puzzle is to determine the unique tiling with those centers. An example of a Spiral Galaxies puzzle and its solution are shown in Figure 1. Figure 1. A Spiral Galaxies puzzle (left) and its solution (right) We will show that the question of whether or not a given Spiral Galaxies puzzle has a solution is NP-complete. To do so, we construct Spiral Galaxies puzzles which correspond to arbitrary Boolean circuits. A circuit will be satisfied, (that is, have a set of inputs which give the desired outputs) if and only if the corresponding puzzle has a solution. Since Satisfiability is the canonical NP-complete problem [4], this will show that Spiral Galaxies puzzles are NP-hard. We complete the proof by showing that a solution to a Spiral Galaxies

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