THE SETUP
Fill every cell with a number from 1 to N (where N = total cells). Every pair of consecutive numbers (k and k+1) must occupy physically adjacent cells — including diagonals. Given clues are fixed. Every cell is visited exactly once.
Critical: diagonal cells are adjacent. An interior cell has up to 8 neighbours. Forgetting diagonals is the single most common reason a path appears stuck when it isn't.
TECHNIQUE 1 — LIST ALL GAPS AND SORT BY SIZE
A "gap" is the stretch of unknown cells between two consecutive given clues. If the clues are 8 and 12, the gap contains values 9, 10, 11 — a gap of size 3.
List every gap:
- Find all given clues and sort them in order.
- For each pair of adjacent clues (A, B), compute gap size = B − A − 1.
- Sort gaps from smallest to largest.
Small gaps (0, 1, 2) are almost always forced. Work them first.
TECHNIQUE 2 — TRACE FORCED PATHS IN SMALL GAPS
Gap = 0: clue A and clue B are directly adjacent. Verify they share a physical neighbour (including diagonal) — if not, something is wrong earlier in the solve.
Gap = 1: one cell must sit between clue A and clue B. That cell must be adjacent (including diagonal) to both. Count the cells adjacent to A AND adjacent to B. If only one such cell exists, place the value there.
Gap = 2: two cells must connect A to B. The first must be adjacent to A; the last must be adjacent to B; and the two must be adjacent to each other. Trace all paths of length 2 from A that can reach B. If only one route exists, place both cells.
For gap = 1 or 2, draw all 8 neighbours of each anchor cell. The intersection is usually 1–2 cells.
TECHNIQUE 3 — CORNER AND EDGE CONSTRAINTS
Corner cells have only 3 neighbours; edge cells have 5. Whatever value sits in a corner, both the value before it (corner − 1) and the value after it (corner + 1) on the chain must fit within those 3 neighbours. Often those constraints narrow the corner to one or two candidate values.
Check every corner cell early. If a nearby clue occupies a corner neighbour, the set of possible corner values is very small.
The two chain endpoints (1 and N) are special: each has only one chain neighbour (2 and N−1 respectively). Place them in corners or edges first if not already given — they have the fewest valid positions.
TECHNIQUE 4 — SHARED CELLS WHEN MULTIPLE ROUTES EXIST
When a gap has more than one valid path, don't guess — instead check whether all valid routes share any cells. If every possible path through a gap passes through the same cell, that cell's value is forced regardless of which route is correct.
This technique is most useful for gaps of size 3–4 where two or three routes exist. Mark the forced cells and move on to other gaps — the ambiguous ones usually resolve once adjacent gaps are filled.
THE SOLVING ORDER
- List all gaps and sort by size (smallest first).
- Place chain endpoints (1 and N) if not given — prefer corners.
- Fill all gaps of size 0 (verify adjacency) and size 1–2 (trace forced paths including diagonals).
- For each newly placed cell, recheck all gaps whose endpoints are now closer — new placements often convert a size-4 gap into two size-1 gaps.
- Apply corner constraints to any corner cells still unknown.
- For remaining gaps: check shared cells across all valid routes.
- Repeat from step 3.
THE BEGINNER TRAP
Tracing the chain continuously from 1 to N without sorting gaps first. This approach forces you to work on the largest gaps when they're least constrained, while the small forced gaps — which would unlock the big ones — sit ignored.
Always sort by gap size. Fill the smallest gaps first. A chain where most of the short gaps are filled looks very different from a chain where you've been chasing the start-to-end narrative — and the short gaps almost always unblock the long ones.