Slant
Draw a diagonal slash in every cell matching the corner numbers, without ever closing a loop.
Basic Game Premise & Rules
MechanicsSlant 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.
- Draw a diagonal line across every single cell in the grid, either / or \.
- A number at a corner intersection (from 0 to 4) indicates exactly how many diagonal lines touch that corner vertex.
- Unnumbered intersections may have any count of touching diagonals.
- No Closed Loops: The diagonal lines can never form a closed loop of any size, anywhere on the board.
- Every board has a unique solution discoverable through spatial and graph-theoretic reasoning.
Rules by Example
See how each rule looks when done correctly versus when violated:
Corner Numbers
Every cell gets a diagonal, / or \. A number at a corner says how many diagonals touch that corner, from 0 to 4.
No Loops
The diagonals never close a loop, small or large.
History & Origins
Origins & EvolutionSlant was created by Nikoli and debuted in *Puzzle Communication Nikoli* #92 in Autumn 2000 under the delightfully playful Japanese title *Gokigen Naname* (ごきげんななめ, 'In a Diagonal Mood').
By constraining slashes across cell diagonals rather than along grid lines, Slant introduced a fresh topological flavor. Simon Tatham included it in his celebrated Portable Puzzle Collection under the name **Slant**. Solvers experience a constant tension: meeting local corner degrees while vigilantly preventing slashes from closing into cycles.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Slant formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Acyclic Subgraphs (Forests) on Grid Graphs
The diagonal segments define edges in a graph G whose vertices are the cell corners. The rule that diagonals never form a closed loop means G must be a **forest** (an acyclic planar graph). If a path already connects vertex u and vertex v, no future diagonal can connect u and v, as it would create a cycle.
Vertex Degree Realization (0 to 4)
Corners hold integers from 0 to 4 indicating incident degrees. A '0' immediately forces all 4 surrounding diagonals to turn away; a '4' forces all 4 diagonals to point inward. Corners of degree 1 and 3 create directional parity chains that propagate diagonally across the board.
NP-Completeness via Planar 3-SAT
Deciding whether a Slant grid has a valid loop-free diagonal completion is NP-complete. The combination of degree realization and cycle avoidance can simulate logic gates and wire crossovers.
References & Further Reading
Bibliography- Nikoli (2000). Puzzle Communication Nikoli #92. Tokyo: Nikoli Publishing.
- Hearn, R. A., & Demaine, E. D. (2009). Games, Puzzles, and Computation. CRC Press.
- Tatham, S. (2004). 'Slant'. Simon Tatham's Portable Puzzle Collection.
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 | Corner 0s, 4s, and simple cycle prevention. | 3 |
| Easy | 5×5 to 6×6 | Boundary corner clues and 3-corner diagonal forces. | 12 |
| Medium | 7×7 to 8×8 | Long diagonal connectivity chains and cycle traps. | 12 |
| Hard | 8×8 to 10×10 | Sparse corners requiring global tree structure deduction. | 12 |
| Expert | 10×10 | Master grids requiring comprehensive graph cycle avoidance. | 11 |
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