Searchlights
Place light beams in white cells so that every numbered black post sees its exact clue count.
Basic Game Premise & Rules
MechanicsSearchlights 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 lights in some white cells.
- Each numbered black cell counts the exact total number of lights it can see along its row and its column.
- Black cells block line of sight; light beams pass through other lights and white cells.
- Lights can shine over other lights, but cannot shine through black barrier cells.
- Every grid has a single unique placement of lights deducible through spatial logic.
History & Origins
Origins & EvolutionSearchlights was developed in Japan as an inverted variation of the classic illumination puzzle. Rather than having light bulbs illuminate white cells (as in Light Up), the black cells themselves act as monitoring stations, counting how many active lights are situated in their direct horizontal and vertical lines of sight.
This inversion changes the mathematical texture: instead of satisfying an art-gallery coverage problem, solvers balance orthogonal sightline intersections, identifying cells where multiple searchlight cones overlap.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Searchlights formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Set Cover & Hitting Sets on Ray Systems
Searchlights can be formulated as an exact Hitting Set problem on intersecting horizontal and vertical rays. Each clue defines a linear equation over binary variables representing lights in visible white cells: ∑_{c ∈ Ray_H(v)} x_c + ∑_{c ∈ Ray_V(v)} x_c = clue(v).
Intersection Multiplicity & Beam Crossings
When two black stations share an intersecting white cell (r, c), placing a light at (r, c) contributes simultaneously to both stations' counts. Solvers use inclusion-exclusion bounds to deduce whether an intersection is required or forbidden.
NP-Completeness
Because it generalizes 0-1 Integer Linear Programming on orthogonal incidence matrices, deciding the solvability of arbitrary Searchlight grids is NP-complete by reduction from 3-SAT.
References & Further Reading
Bibliography- Demaine, E. D., & Hearn, R. A. (2009). Games, Puzzles, and Computation. CRC Press.
- Seta, T. (2002). 'The Complexity of Japanese Pencil Puzzles'. ISAAC 2002.
- Garey, M. R., & Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness.
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 | Single-beam monitors and obvious non-overlapping white lines. | 3 |
| Easy | 5×5 to 6×6 | Double-monitor intersections and corner zero exclusions. | 12 |
| Medium | 7×7 to 8×8 | Dense sightlines where beams cross multiple stations. | 12 |
| Hard | 8×8 to 10×10 | Sparse clues forcing global multi-station intersection analysis. | 10 |
| Expert | 10×10 | Master grids requiring comprehensive 0-1 linear inequality solving. | 5 |
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