Light Up
Place light bulbs to illuminate every white cell without any two shining on each other.
Basic Game Premise & Rules
MechanicsLight Up 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.
- Place light bulbs in white cells so that every white cell is illuminated.
- Each light bulb shines rays horizontally and vertically across white cells until blocked by a black cell or the grid edge.
- No Two Bulbs May Clash: No two bulbs can shine on each other (they cannot share a row or column without at least one black barrier between them).
- Clues on Black Cells: A number on a black cell indicates exactly how many bulbs must be placed in orthogonally adjacent white cells (0, 1, 2, 3, or 4).
- Unnumbered black cells may have any count of adjacent light bulbs.
Rules by Example
See how each rule looks when done correctly versus when violated:
Illumination & Light Rays
Each light bulb shines light horizontally and vertically until blocked by a black cell or the grid edge. Every white cell must be illuminated.
No Mutual Illumination (Bulb Conflicts)
No two light bulbs may shine on each other. Two bulbs cannot share a row or column unless there is a black wall between them.
Numbered Black Cells
A number on a black cell indicates the exact count of light bulbs in its orthogonally adjacent white cells. Unnumbered black cells have no count restriction.
History & Origins
Origins & EvolutionLight Up was invented by Nikoli and debuted in *Puzzle Communication Nikoli* #95 in 2001 under the title *Akari* (あかり, 'Light'). It was also called *Bijutsukan* ('Art Gallery'), a direct homage to the famous **Art Gallery Problem** in discrete mathematics formulated by Victor Klee in 1973.
Light Up became an instantaneous international classic. The brilliance of its design lies in its luminous clarity: every placed bulb sends visible yellow rays streaming across corridors, immediately illuminating large swaths of the board while strictly forbidding any other bulb from encroaching on its line of sight.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Light Up formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
The Art Gallery Problem on Orthogonal Grids
Light Up directly formalizes the Art Gallery Problem on orthogonal polygons with rectilinear barriers. While classic art gallery problems ask for the minimum number of guards to illuminate a polygon, Light Up adds the mutually independent guard condition: no two guards may have line of sight to each other.
NP-Completeness via Circuit SAT
In 2005, Laura McPhail published a proof proving that deciding whether a Light Up board has a valid bulb placement is NP-complete. By constructing logic gates (AND, OR, NOT) using corridors of light and black blocks, McPhail showed that any Boolean circuit can be simulated on a Light Up grid.
2-SAT Conflict Clauses & Corner Traps
The mutual non-illumination rule generates pure 2-SAT clauses: for any two white cells u, v sharing a row or column without an intervening black cell, (¬b_u ∨ ¬b_v). Clues on black cells generate cardinality constraints, allowing fast propagation of forced placements.
References & Further Reading
Bibliography- McPhail, L. (2005). 'Light Up is NP-complete'. Technical Report, Department of Computer Science.
- Nikoli (2001). Puzzle Communication Nikoli #95. Tokyo: Nikoli Publishing.
- O'Rourke, J. (1987). Art Gallery Theorems and Algorithms. Oxford University Press. ISBN: 978-0195039658.
- Chvátal, V. (1975). 'A combinatorial theorem in plane geometry'. Journal of Combinatorial Theory, Series B, 18(1), 39–41.
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 boards demonstrating 4-cell and 0-cell corner deductions. | 3 |
| Easy | 5×5 to 6×6 | Diagonal 3-clues, forced bulb rays, and unlit cell covers. | 12 |
| Medium | 7×7 to 8×8 | Interlocking light corridors and isolated corner illuminations. | 12 |
| Hard | 8×8 to 10×10 | Multi-bulb clash prevention and distant dark cell coverage. | 12 |
| Expert | 10×10 | Complex architectural layouts with sparse clues demanding deep lookahead. | 11 |
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