STEP 1 — SORT CLUES AND COMPUTE ALL GAPS
List every given clue in ascending order. The path runs from 1 to the total cell count without skipping — so every pair of consecutive clues defines a gap: gap = (higher clue − lower clue − 1). A gap of 0 means the two clues are adjacent; a gap of 1 means exactly one cell sits between them.
Sort gaps from smallest to largest. You will fill them in that order.
STEP 2 — FILL GAP-0 (ADJACENT CLUES)
Two clues with a gap of 0 must be in neighbouring cells. Check whether they already are — if not, the puzzle has an error. If they are adjacent, mark that edge as confirmed. This often locks one clue cell's exit direction and forces the next segment.
STEP 3 — EXPLOIT DEAD-ENDS AND CORNER CELLS
Any cell with only one free neighbour that the path must pass through is a dead-end: the path enters and exits through the same bottleneck. Corner cells on a square grid have exactly two neighbours — if the path must visit a corner, both entry and exit are fixed.
Look for these before working on large gaps. A single forced corner often chains into several forced steps without any counting.
STEP 4 — FILL SMALL GAPS (1–3) VIA SHARED NEIGHBOURS
For each gap-1 pair: the missing cell must be a common neighbour of both clue cells. List the neighbours of each clue, find the intersection — if it's a single cell, place the intermediate number. For gap-2: the two missing cells must form a path between the clues; enumerate short routes and keep only those consistent with the grid.
Even if multiple routes exist for a gap, any cell that appears in every valid route is forced. Place it.
STEP 5 — USE CONNECTIVITY TO ELIMINATE ROUTES
The path visits every cell exactly once and cannot branch. If a candidate route for one gap would cut off a pocket of unvisited cells from the rest of the grid — leaving cells unreachable — that route is illegal. Eliminate it.
After placing a new segment, re-check the grid for isolated groups. A pocket of N cells with only one entry point forces the path through that entry in a specific direction.
SOLVING ORDER
- Sort clues, compute all gaps.
- Confirm or derive gap-0 adjacencies.
- Trace dead-ends and corner cells.
- Fill gap-1, then gap-2, then gap-3 via shared neighbours.
- Use connectivity to prune.
- Repeat until solved.
For beginner puzzles the first three steps typically place 40–60% of cells before any neighbour-counting is needed.
BEGINNER TRAP — FORGETTING THE PATH VISITS EVERY CELL
The path doesn't just connect the given clues — it must pass through every empty cell in the grid. A route that skips a block of cells is always wrong, even if it connects the clues correctly. When a candidate segment looks clean, count the cells it leaves unvisited and check whether the remaining gaps can cover them.