THE CORE RULE
Consecutive numbers must occupy orthogonally adjacent cells — sharing an edge, not a corner. Cell N must be directly next to cell N−1 and cell N+1. The path visits every cell exactly once and forms a single connected chain from the lowest to the highest number in the grid.
Start and end cells are pre-marked. Clue cells are fixed. The solution is unique — deduction, not search.
THE RECTANGULAR GRID IS NOT UNIFORM
Interior cells have 4 free neighbours. Edge cells have 3. Corner cells have 2. This asymmetry is the source of most forced moves in Square Maze. A path that must pass through a corner cell — without starting or ending there — must use both of the corner's two neighbours as entry and exit. That severely restricts which numbers can sit in the corner.
Scan corners before scanning the interior. A clue one or two steps from a corner frequently forces 3–6 cells without any gap counting.
GAPS BETWEEN CLUES ARE YOUR UNITS OF WORK
Between two consecutive clues, a fixed number of intermediate cells must be threaded. If clue A = 5 and clue B = 9, exactly three cells (6, 7, 8) must fit between their grid positions. Count how many cells are reachable between A and B without crossing other clue positions. If only one spatial route accommodates the right count, the gap is forced.
Small gaps are almost always forced. List all gaps at the start of every solve and sort by size — smallest first.
DEAD-END CASCADES
A dead-end is a cell with only one free orthogonal neighbour. The path must both enter and exit every non-endpoint cell, so a dead-end in the middle of the path is impossible. When a cell becomes a near-dead-end (one free neighbour), the number in that neighbour is forced to be either the predecessor or successor of the cell's own value.
Dead-end cascades are the most productive technique. After every placement, scan the surrounding cells for new dead-ends. One forced move frequently creates two or three more in sequence.
CONNECTIVITY CHECK
As the path fills in, verify that the remaining empty cells still form one connected region. If placing a number would cut the empty region into two disconnected parts — and the path still needs to visit both — that placement is illegal.
Connectivity is especially powerful in narrow corridors and near the path's current endpoint. A pocket of K unvisited cells with a single entry point forces the path through that entry in a specific direction.
THE SOLVING LOOP
- Examine corner and edge cells — locate any with forced entry-exit pairs.
- List all gaps between clue pairs. Sort smallest first. Force any gap-0 or gap-1 segments immediately.
- After each placement, scan for dead-end cells and cascade them.
- Apply the connectivity check before committing any uncertain placement.
- Move to the next smallest unfilled gap and repeat.
THE BEGINNER MISTAKE
Tracing the path forward from cell 1, choosing direction by intuition. The interior quickly branches into too many options to track. Instead, start from clue cells and corners — the most constrained positions — and work inward. Clues and boundaries reduce possibilities far faster than forward-tracing from the start.