THE CORE RULE
Adjacent cells on the path must hold consecutive numbers. Cell with value N must be directly next to the cell with value N−1 and the cell with value N+1. "Adjacent" means sharing an edge (for square grids) or following the grid topology (circular or hex).
Start and end cells are pre-marked. Some numbered cells are given as clues and cannot be moved. The path is unique — deduction, not search.
GAPS BETWEEN CLUES ARE YOUR UNITS OF WORK
Between two adjacent given clues, the path must pass through a fixed number of intermediate cells. If clue A = 4 and the next clue B = 9, then exactly four cells (5, 6, 7, 8) must fit between their positions.
Count how many grid cells are reachable between A and B without revisiting other fixed clue positions. If only one spatial route accommodates the right number of intermediate cells, the path through that gap is forced.
Small gaps are almost always forced. Start every solve by listing all gaps and sorting by size — smallest first.
CORRIDORS AND DEAD ENDS
In square mazes, the path can only enter or leave a cell via its four cardinal neighbours. If a cell has only two open neighbours (the others being boundaries or already-placed cells), then those two neighbours are the only possible entry and exit points — the path through that cell is completely determined.
True dead-end cells (only one open neighbour) can only be either the start, the end, or a forced turnaround. If a dead-end cell must be visited (because it falls in a required gap), the path must enter and immediately exit through the same opening — which constrains the sequence of numbers nearby.
CONNECTIVITY CHECK
As you place cells, check that the unplaced region remains connected. If placing a cell would split the remaining empty area into two disconnected regions — and the path still needs to visit both — that placement is invalid.
This check is more powerful than it sounds. It eliminates entire families of candidate paths without enumerating them, especially in mazes with narrow passages.
THE SOLVING LOOP
- List all gaps between adjacent clue pairs. Note gap size (cells needed) and spatial distance between the two anchor clues.
- For the smallest gap: enumerate routes between the two anchors. If only one route fits the cell count, place those numbers.
- After placing, recheck dead-end and two-neighbour cells. Any that now have forced entry-exit pairs can be placed immediately.
- Apply the connectivity check to any uncertain placement before committing it.
- Repeat until the path is complete.
THE BEGINNER MISTAKE
Tracing the path from cell 1 forward, choosing directions by intuition. The path quickly branches into too many options to track mentally. Work from given clues inward, not from the start outward — clues reduce possibilities far more efficiently than forward-tracing.