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]!



Paper star battles
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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.

Title						CELL					DATE			STAR			PUZZLE
Cairo		 				pentagonal				06/03/2021		☆				[star-battle/images/22-cairo-star-battle.png|star-battle/images/22-cairo-star-battle.png]
Fractal 					square (variable size)	21/12/2020		☆☆				[star-battle/images/25-fractal-star-battle.png|star-battle/images/25-fractal-star-battle.png]
Basket weave 				rectangle				12/12/2020		☆				[star-battle/images/19-basket-weave-star-battle.png|star-battle/images/19-basket-weave-star-battle.png]
Half-hex					hexagon and trapezium	05/12/2020		☆				[star-battle/images/18-half-hex-star-battle.png|star-battle/images/18-half-hex-star-battle.png]
Kite and Dart				kite and dart			28/11/2020		☆				[star-battle/images/17-kite-and-dart-star-battle.png|star-battle/images/17-kite-and-dart-star-battle.png]
Good Morganing 				ellipse intersections	21/11/2020		☆☆				[star-battle/images/16-good-morganing-ellipse.png|star-battle/images/16-good-morganing-ellipse.png]
Good Morganing				circle intersections	20/11/2020		☆☆				[star-battle/images/15-good-morganing-circle.png|star-battle/images/15-good-morganing-circle.png]
Square Prism (in space)		square					10/10/2020		☆☆				[star-battle/images/14-star-battle-square-prism.png|star-battle/images/14-star-battle-square-prism.png]
Cube (in space)				square					04/10/2020		☆☆				[star-battle/images/13-cube-star-battle.png|star-battle/images/13-cube-star-battle.png]
Tetrahedron (in space)		triangle				12/09/2020		☆				[star-battle/images/12-tetrahedral-star-battle.png|star-battle/images/12-tetrahedral-star-battle.png]
Ammann-Beenker				lozenge and square		04/09/2020		☆				[star-battle/images/11-ammann-beenker-star-battle.png|star-battle/images/11-ammann-beenker-star-battle.png]
Moebius (in space)			square					29/08/2020		☆☆				[star-battle/images/10-moebius-star-battle.png|star-battle/images/10-moebius-star-battle.png]
Penrose						rhombus					11/08/2020		☆				[star-battle/images/09-penrose-star-battle.png|star-battle/images/09-penrose-star-battle.png]
Icosahedron (in space)		triangule				09/08/2020		☆☆				[star-battle/images/08-icosahedron-star-battle.png|star-battle/images/08-icosahedron-star-battle.png]
Polar						square (perspective)	01/08/2020		☆☆				[star-battle/images/07-polar-star-battle.png|star-battle/images/07-polar-star-battle.png]
Hyperbolic					square (deformed)		12/07/2020		☆				[star-battle/images/06-hyperbolic-star-battle.png|star-battle/images/06-hyperbolic-star-battle.png]
Hyperbolic					square (deformed)		11/07/2020		☆				[star-battle/images/05-hyperbolic-star-battle.png|star-battle/images/05-hyperbolic-star-battle.png]
Cube (in space)				square					19/06/2020		☆				[star-battle/images/04-cube-star-battle.png|star-battle/images/04-cube-star-battle.png]
Triangular					triangle				18/06/2020		☆☆				[star-battle/images/03-triangular-star-battle.png|star-battle/images/03-triangular-star-battle.png]
Triangular					triangle				17/06/2020		☆				[star-battle/images/02-triangular-star-battle.png|star-battle/images/02-triangular-star-battle.png]



Interactive star battles
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${ScriptLoad("projects/star-battle/hyper-loader.js"),""}

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

Title						CELL					DATE			STAR	PUZZLE
Cairo		 				pentagonal				06/03/2021		☆		[star-battle/images/22-cairo-star-battle.png|star-battle.html?c=-sb+-zoom+.6&1=22-cairo.lev]
Kite and Dart				kite and dart			28/11/2020		☆		[star-battle/images/17-kite-and-dart-star-battle.png|star-battle.html?c=-sb+-zoom+.6&1=17-kite-and-dart.lev]
Cube (in space)				square					04/10/2020		☆☆		[star-battle/images/13-cube-star-battle.png|star-battle.html?c=-sb&1=starbattle-cube-13.lev]
Icosahedron (in space)		triangle				09/08/2020		☆☆		[star-battle/images/08-icosahedron-star-battle.png|star-battle.html?c=-sb&1=starbattle-icosahedron.lev]
Cube (in space)				square					19/06/2020		☆		[star-battle/images/04-cube-star-battle.png|star-battle.html?c=-sb&1=starbattle-square.lev]
Triangular					triangle				18/06/2020		☆☆		[star-battle/images/03-triangular-star-battle.png|star-battle.html?c=-sb+-zoom+.5&1=starbattle-triangle2.lev]
Triangular					triangle				17/06/2020		☆		[star-battle/images/02-triangular-star-battle.png|star-battle.html?c=-sb+-zoom+.6&1=starbattle-triangle1.lev]



Controls
====================

Placing //stars//
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To place a star, [[click]] on an empty //cell//.
To remove a star, [[click]] on it. [[drag]] around to remove many.

Placing marks
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To place a mark, [[auxclick]] on an empty //cell//. Drag around to place many marks.
To remove a mark, [[auxclick]] on it. [[auxdrag]] around to remove many.

Panning
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To rotate the puzzle, [[wheel-click]]. The position of the mouse around the center determines the direction of rotation.
You may also pan with the ${Hyper("ArrowKeys/Arrows")}.

Checking for errors
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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]].

Editing & Advanced
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Open the options menu by pressing [[v]]. You can use this to make your own puzzles and edit the geometry!




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



Star battle rules
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Standard star battle puzzles
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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.


Geometric star battle puzzles
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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//.


How to make geometric star battle puzzles
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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!


Forward design approach
========================

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,IHNN @@.


Catalogue of interesting //small regions//
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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.

Adjacent constraints
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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. 

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


Grouping //lines// or //regions//
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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.


Divide and conquer method
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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.

Backward design approach
=========================

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 //constellation//
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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.