THE SETUP
A Square Maze is a rectangular grid of N cells. Every integer from 1 to N is placed exactly once, one per cell. The rule: consecutive integers must occupy orthogonally adjacent cells — sharing an edge, not just a corner. The result is a single Hamiltonian path snaking through the entire grid from the smallest number to the largest.
Some integers are given as fixed clues. The gaps between them must be filled in. Unlike Hidato (which uses all eight directions), Square Maze uses only four. That restriction makes the grid feel tighter: paths back up faster, dead-ends appear sooner, and the corners of the rectangle are constrained from the very first move.
CORNERS ARE THE STARTING POINT
The rectangular grid is not spatially uniform. Interior cells have four neighbours. Edge cells have three. Corner cells have exactly two. A corner cell that isn't the path's start or end must be both entered and exited — through just two available neighbours. The cell immediately before it and the cell immediately after it in the sequence must each occupy one of those two neighbours.
In practice, scan every corner cell before computing a single gap. If any clue number is one or two steps from a corner, the path through that corner is often completely determined — three or four cells placed without any counting.
Corner forcing works on every rectangular grid regardless of size. Even a 10×10 grid with no clues near the interior yields immediately once you examine its four corners.
GAP ANALYSIS — THE SECONDARY TOOL
After exhausting corner and edge constraints, work the gaps. A gap between two consecutive given clues tells you exactly how many intermediate cells the path must thread between their positions. A gap of 1 means the missing cell must be a common neighbour of both anchors — often just one candidate on a constrained grid.
Sort all gaps smallest-first. Gaps of 0 (adjacent clues) are trivially forced. Gaps of 1–3 usually have very few routing options near walls or filled cells. Work them in order; each placement creates new dead-ends that drive the next.
DEAD-END CASCADES
A dead-end is a cell with only one free orthogonal neighbour. Any non-endpoint cell in that state forces the neighbour to hold either its predecessor or successor in the sequence. After every placement, scan the surrounding cells for near-dead-ends — one forced move typically creates two or three more in sequence.
This cascade is the fastest technique in Square Maze. A single corner-forced placement near the edge can propagate through an entire row before any gap analysis is needed.
HOW IT COMPARES TO OTHER MAZE VARIANTS
Square Maze is the four-direction variant of a family of consecutive path puzzles. Circular Maze uses the same four-direction rule but on concentric rings — the inner rings replace corners as the most constrained area. Hex Mazes use six directions on a hexagonal grid, which loosens routing significantly. Hidato (Starlink) uses eight directions on a square grid, trading the tighter path constraints for diagonal routing freedom.
Among the four variants, Square Maze is the most spatially transparent: every player immediately understands which cells are adjacent on a rectangular grid. There's no topology to learn. The puzzle is pure path logic on a familiar canvas.
THE SOLVE ORDER
- Examine all four corners — fill any forced entry-exit pairs.
- List gaps between consecutive clues. Sort smallest-first. Force gap-0 and gap-1 segments immediately.
- After each placement, cascade dead-ends — scan for cells with one free neighbour and resolve them.
- Verify connectivity: no candidate route should cut off remaining empty cells into an isolated pocket.
- Move to the next gap and repeat.
Most beginner Square Mazes resolve entirely from step 1 and step 3. Gap analysis only comes in for grids where all clues sit in the interior away from boundaries.