Galaxies
Partition the grid into rotational 180° symmetric galaxies, each centered on a single dot.
Basic Game Premise & Rules
MechanicsGalaxies is an untimed, deterministic pencil-and-paper logic puzzle played on a grid. Every valid puzzle has exactly one unique solution that can be discovered through rigorous deductive reasoning alone, without trial and error or guessing.
- Draw borders along grid lines to split the entire grid into regions ('galaxies').
- Every galaxy must contain exactly one dot (the galactic core).
- 180° Rotational Symmetry: Each galaxy must possess 2-fold rotational symmetry about its dot: rotating the galaxy 180° around its dot leaves the shape unchanged.
- Dots may sit in the exact centre of a cell, on the grid line between two cells, or at the intersection of four cells.
- No border can pass through a dot.
Rules by Example
See how each rule looks when done correctly versus when violated:
One Dot in Every Region
Borders split the grid into regions, and every region holds exactly one dot. A dot may sit in a cell, on the line between two cells, or where four cells meet; no border passes through it.
Half-Turn Symmetry
Turn a region half a turn (180°) about its dot and it lands exactly on itself: the dot is the region's centre.
History & Origins
Origins & EvolutionGalaxies was invented by Nikoli and debuted in *Puzzle Communication Nikoli* #96 in Autumn 2001 under the title *Tentai Show* (天体ショー, 'Astronomical Show').
It was widely hailed as an immediate masterpiece of visual logic: the board resembles a telescope peering into deep space, with celestial dots scattered across a dark sky. Solvers trace rotating spiral arms around each nucleus, discovering that every cell belongs to a unique galaxy. It was introduced to computer gamers worldwide through Simon Tatham's puzzle collection as **Galaxies**.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Galaxies formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Point Reflection Groups (C2 Symmetry)
In group theory, each galaxy region R ⊂ ℤ² is invariant under the point-reflection group C_2 centered at (x_0, y_0): for every cell (r, c) ∈ R, the reflection (2x_0 - r, 2y_0 - c) must also belong to R. This enforces pairwise cell couplings: capturing cell A immediately mandates capturing its antipodal partner A'.
NP-Completeness via Planar 3-SAT
In 2002, Erich Friedman proved that Tentai Show (Galaxies) is NP-complete. By configuring dots so that symmetric arms act as Boolean wires and clause arbiters, Friedman proved that deciding whether a grid can be partitioned into 180°-symmetric galaxies is computationally intractable.
Antipodal Conflict & Voronoi Duals
If two dots claim ownership of overlapping antipodal pairs, mutual exclusion applies. Galaxies can be viewed as an integer lattice discrete Voronoi partition under reflectional distance metrics.
References & Further Reading
Bibliography- Friedman, E. (2002). 'Spiral Galaxies Puzzles are NP-complete'. Stetson University Technical Reports.
- Nikoli (2001). Puzzle Communication Nikoli #96. Tokyo: Nikoli Publishing.
- Tatham, S. (2004). 'Galaxies'. Simon Tatham's Portable Puzzle Collection.
Free Printable PDF Booklet
Download our beautifully formatted, high-resolution A4 puzzle booklet. Generated directly from the handcrafted levels on AZTest, it contains 50 puzzles ordered from gentle introductory boards up to master-level challenges, followed by complete solutions at the back.
| Difficulty Stage | Grid Size | Description | Puzzles |
|---|---|---|---|
| Intro | 5×5 | Small symmetric bars and crosses centered on cell centers. | 3 |
| Easy | 5×5 to 6×6 | Edge dots and vertex intersections with clear reflection partners. | 12 |
| Medium | 7×7 to 8×8 | Interlocking S-shapes and L-shaped spiral arms. | 12 |
| Hard | 8×8 to 10×10 | Dense constellations where galaxies wrap around adjacent cores. | 12 |
| Expert | 10×10 | Grandmaster cosmos requiring global geometric coordination. | 11 |
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