Galaxies (Tentai Show)

Galaxies, also known as Tentai Show or Spiral Galaxies, is a grid puzzle where the player divides a rectangular grid into regions, each containing exactly one dot (the "galaxy center"). Every region must be rotationally symmetric around its center dot — if you rotate the region 180 degrees around the dot, it looks exactly the same. The dots may be placed at cell centers, on cell edges, or at cell intersections, producing regions of different shapes and sizes.

Rules

1. The grid contains dots placed at various positions (cell centers, edges, or intersections).
2. Divide the entire grid into regions so that every cell belongs to exactly one region.
3. Each region contains exactly one dot.
4. Each region must have 180-degree rotational symmetry around its dot. That is, for every cell in the region, the cell at the opposite side of the dot (at equal distance, reflected through the dot) must also be in the region.

Strategies

• Dot Position Determines Minimum Size: A dot at a cell center requires at least that one cell. A dot on an edge between two cells requires at least those two cells. A dot at an intersection requires at least the four surrounding cells.
• Symmetry Constraint: For every cell you assign to a region, you must also assign its rotational counterpart through the region's dot. This means regions always have an even number of cells (unless the dot is at a cell center, giving an odd count).
• Boundary Propagation: If a cell on the edge of the grid belongs to a region, its symmetric counterpart must also be within the grid. Cells near edges often have restricted options.
• Neighbor Exclusion: If two dots are close together, their regions cannot overlap. Determine the boundary between them by considering which cells each dot can claim while maintaining symmetry.
• Expansion from Dot: Start from each dot and see which cells it must claim. If a dot is near a corner, the symmetry constraint heavily limits the region shape.
• Remaining Space: As regions are determined, remaining unassigned cells must belong to nearby dots. If only one dot can symmetrically claim a cell, that assignment is forced.

Galaxies was invented by Nikoli, the Japanese puzzle publisher, and first appeared in their puzzle magazines in 2001 under the Japanese name "Tentai Show" (which roughly translates to "astronomical show" or "celestial show," evoking the image of spiral galaxies). The English name "Galaxies" or "Spiral Galaxies" was adopted for international audiences, referencing the visual resemblance of the symmetric regions to spiral galaxy shapes.

The puzzle is a beautiful example of Nikoli's design philosophy: a single elegant constraint (rotational symmetry) produces rich logical challenges. The placement of dots at edges and intersections (not just cell centers) was a particularly clever design choice, as it creates regions of varied sizes and shapes that interlock in complex ways.

Galaxies has been featured in the World Puzzle Championship and is a favorite among competitive puzzle solvers for its visual and spatial reasoning demands. Unlike many grid puzzles that rely on numbers and arithmetic, Galaxies is purely geometric, requiring the solver to think about symmetry and shape.

The puzzle was included in Simon Tatham's Portable Puzzle Collection under the name "Galaxies," which helped bring it to a broader audience. It has since appeared in various puzzle apps and online puzzle platforms. The puzzle has also attracted interest from computer science researchers, as the computational complexity of solving general Galaxies instances has been studied — the problem is NP-complete, though well-designed puzzles are solvable through pure deduction.