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Star Battle geometries

Updated on (created ) — Filed under: Geometry Guide Original piece
Cairo22-cairo-star-battle.png Fractal25-fractal-star-battle.png Basket weave19-basket-weave-star-battle.png

Possible future puzzles are listed under More Ideas. Excited about a particular idea? Do !

What if you could play Star Battle on different geometries? Now you can!.

Geometric star battles include triangular, penrose, Amman-beenker and hyperbolic tilings of the plane, star battle in polar corrdinates (polar star battle) and in three dimensions, folding cubes, icosahedron, and möbius strip (star battles in space)! The puzzles come in varying difficulties. One star puzzles tend to be easier, and two star puzzles a bit harder, but not always! Choose between old-school Paper star battles and the newest Interactive star battles, or enjoy a couple Hybrid star battles!

To play these, all you need is pen and paper! Or, if you prefer, the flood-fill tool will work on every puzzle, e.g using MSPaint or other image editing software. Shall a star battle be named in space, then part of the challenge is folding the puzzle either in your head or an actual paper version.

Cairo22-cairo-star-battle.png Fractal25-fractal-star-battle.png Basket weave19-basket-weave-star-battle.png Half-hex18-half-hex-star-battle.png Kite and Dart17-kite-and-dart-star-battle.png Ellipses16-good-morganing-ellipse.png Circles15-good-morganing-circle.png Square Prism14-star-battle-square-prism.png Cube13-cube-star-battle.png Tetrahedron12-tetrahedral-star-battle.png Ammann-Beenker11-ammann-beenker-star-battle.png Moebius10-moebius-star-battle.png Penrose09-penrose-star-battle.png Icosahedron08-icosahedron-star-battle.png Polar07-polar-star-battle.png Hyperbolic06-hyperbolic-star-battle.png Hyperbolic05-hyperbolic-star-battle.png Cube04-cube-star-battle.png Triangular03-triangular-star-battle.png Triangular02-triangular-star-battle.png

Pick one of the puzzles below to launch the interactive 3D star battle player in your browser, thanks to ZenoRogue!

Cairo22-cairo-star-battle.png Kite and Dart17-kite-and-dart-star-battle.png Cube13-cube-star-battle.png Icosahedron08-icosahedron-star-battle.png Cube04-cube-star-battle.png Triangular03-triangular-star-battle.png Triangular02-triangular-star-battle.png

To place a star, (tap) or (click) on an empty cell. To remove a star, (tap) or (click) on it. (tap and drag) or (click and drag) around to remove many.

To place a mark, (right click) on an empty cell. Drag around to place many marks. To remove a mark, (right click) on it. (right click and drag) around to remove many.

To rotate the puzzle, (click the mouse wheel button). The position of the mouse around the center determines the direction of rotation. You may also pan with the .

To check for errors press E: the regions will become white (correct number of stars), red (too many stars) or blue (missing stars). To return to the puzzle, press R.

Open the options menu by pressing V. You can use this to make your own puzzles and edit the geometry!

Hybrid star battles are a mix of star battle and other puzzle genres, such as Galaxies or Aquarium: check the Paper Puzzle Trove!

Standard star battle puzzles. ===========================~

In standard star battle puzzles, you must deduce which cells in the (square) grid to place a star on, so that every line and region have a set number of stars (1 star: easy, 2 stars: medium, more stars: hard). Importantly, no two stars may touch each other diagonally, because they may not share vertices.

In star battle puzzles using different geometries, the rules are identical. However, lines may bend or wrap-around, there may be more than two lines per cell, and the number of cells per vertex / vertices per cell may differ, even locally.

  • Planar square grid: 2 lines per cell, (1, 2 or) 4 cells per vertex, (3, 5 or) 8 adjacent cells.
  • Cube square grid: 3 lines per cell, 3 or 4 cells per vertex, 7 or 8 adjacent cells.
  • Planar triangular grid: 3 lines per cell, (2, 3 or) 6 cells per vertex, (6, 7 or) 12 adjacent cells.
  • Planar Penrose grid: 2 lines per cell, 3 to 5 (?) cells per vertex, 8 to 11 (?) adjacent cells.
  • Planar Amman-Beenker grid: 2 lines per cell, (…) 3 to 8 cells per vertex, (…) 8 to 12 adjacent cells.

You found an interesting grid geometry for a star battle puzzle! But… how do you make a puzzle?

At least two approaches can be distinguished, named forward or backward design. The forward approach is recommended for more interesting puzzles, while the backward approach can be a useful starting point or complement, especially in unfamiliar geometries. Both approaches can be combined, by e.g. forward designing to a desired solution.

Finally, you'll need some tools to save you time!

Starting with a blank slate, forward design involves a placing a small set of clues then partially solving the puzzle by exhausting all new deductions implied by those clues. Iterate until the puzzle is complete.

Forward design applies to all paper puzzle types. Below you find some specific star-battle methods, suggested by Deusovi and Jeffrey Bardon (IHNN).

Analyse all possible very small regions, e.g with 7 cells or less. In many cases, there are only a handful of compatible positions in those regions where +2 stars can be placed.

Often, these small regions will constrain any adjacent regions, so by placing them strategically, larger adjacent regions can be constrained quickly. This works even if placing a single star per region.

Also, some small regions may completely exclude a line or exclude one of several lines.

Count the number of regions fully contained in a set of lines and how many stars would need to be placed in those regions. How many stars remain to be placed in those lines? If none, any remaining regions partially contained in those lines will be empty. If negative, you made a mistake. Otherwise you can use that information to constrain the non-overlapping portion of those regions.

Often, combined regions can do interesting things that they don't do alone.

You split the grid into macro regions with a large but fixed number of stars. Then you split those macro regions incrementally until you arrive at the desired regions with the standard number of stars.

The backward approach starts with the solution and attempts to make a puzzle around it. If used in isolation, it may lead to less interesting puzzles. Luckily, it can also be used to give a global direction to a forward design process.

Specifically for star battle, this approach has two steps:

  • finding a constellation (easy)
  • drawing a set of regions with a unique solution (hard)

Let r be the number of regions and n the number of stars per region. In total you'll need s = r x n stars.

Finding a star battle constellation means filling the grid with a set of s stars such that no starred cells share any vertices, and no more than n stars per line. Different geometries impose different rules on which vertices are shared, and on what constitutes a line.

First ensure the grid is not too small, so that it is not impossible to find a constellation. Start by counting s, the total required number of stars. Then notice how many cells are adjacent to each cell (there may be a couple different cases at the grid boundary), and try to discount any overlaps. Do s stars fit the grid? If not, keep enlarging the grid by one unit until they fit.

Then, with a grid in place, the following two imperfect methods are are sometimes useful for finding a constellation.

Usually the grid has some symmetries that can be used. For instance, shall there be a m-fold rotational symmetry around a central point, often you can place a starting star anywhere away from the center, and then place m - 1 star copies at the symmetric points.

Place a starting star anywhere, and mark a ring of all its adjacent cells. Then place a second star outside and adjacent to the ring, as close as possible to the starting star, but on a different line. In a square grid, the second star is now a knight's move away.

Place more stars following the same pattern, until either you wrap-around a boundary and finish the constellation or you must stop. If you must stop, try to place a secondary starting star also adjacent to the starting ring and continue from there.

Finding a valid constellation is relatively easy, and with a single constellation you can make a huge number of star battle puzzles, by drawing different sets of regions. However, ensuring that the puzzle has a unique solution can be very time consuming. Here's one method.

  • Place a valid constellation on the grid, with r x n stars.
  • Draw r-1 small uninteresting regions to confine n x (r-1) stars, and name these regions crystals. Thus only n stars remain outside any crystal, in open territory.
  • The smaller the initial crystals you pick, the easier it is to verify uniqueness. So change the crystals manually now until you ensure a unique solution. Then, in the last step, you'll transform a boring unique solution into an interesting unique solution.
  • Pick one of the r-1 crystals. Grow it by one new cell into open territory where you can quickly check that no star from the crystal could be moved to the new cell (i.e., the new solution is still unique). Repeat this step until you're happy.

For simple square grids, check the handy puzz.link star battle editor.

All interactive star battles are editable, and you can even change the geometry, thanks to ZenoRogue's RogueViz non-Euclidean geometry engine!

A list ideas for future puzzles, for self-reference. If you make one of these, please let me know - happy to add it to the list (with proper credit).

  • Toroidal star battle, with self-intersecting lines. stars are placed at intersections (Thinky Puzzles server).
  • binary star battle 1: two types of stars, heavy and light. Light stars are alone in a line, heavy stars are together;
  • binary star battle 2: two types of stars, blue and red. Blue stars on every line, red stars on every region. stars cannot be adjacent;
  • log spiral grid - current 2 star constellation with too many stars, possibly tedious;
  • liar star battle - where exactly one of the regions contains no stars;
  • starrykabe - the location of the stars is actually the starting point for a nurikabe puzzle (the number assignment would be based either on order of placement of stars ot their coordinates);
  • origami star battle - this would require two orthogonal folds plus transparent paper. Folding could erase and create new regions, and reduce the number of stars.
  • HEGANOX - a laser game, using the half-hex grid

Star Battle geometries by Pedro, 2020-2022.

Special thanks to ZenoRogue for adapting his brilliant RogueViz non-Euclidean geometry engine, for star battles!

Thanks to Skymoo, Portponky, Deusovi, Jack Lance, Velleic and Katelyn Delta, and others in the Thinky Puzzles server, especially for important ideas, useful comments, mistake detection and trying out the triangular puzzles.

Thanks to ZenoRogue for making great tools to explore the hyperbolic plane and to Elyot Grant for inpiring discussions.

Thanks to Portponky and ZenoRogue for help in clarifying the puzzle rules.

Thanks to IceWing for helping perfecting the cube. Thanks to Katelyn Delta and Zachary Barbanell (TheZachMan) for help with the icosahedron fold lines.

Thanks to Jonah Ostroff (softfro) for finding an unintended solution, now corrected.

Thanks to Deusovi, Portponky, Karoo and Arvi Teikari (Hempuli) for helpful comments and finding an unintended solution.

Thanks to Deusovi and Pancelor for help in clarifying the rules and solving the puzzle!

Thanks to Ethan Clark (EPGA Studios) for suggesting the ellipses!