THE SETUP
A Number Maze grid contains N cells — square, circular, or hexagonal depending on the variant. Some cells contain given numbers. Your task: fill every remaining cell with an integer so that consecutive integers occupy orthogonally adjacent cells (sharing an edge, not just a corner).
The result is a single Hamiltonian path — a route through every cell exactly once, numbered 1 to N in order. Given clues anchor the path at fixed points; the gaps between them must be bridged by exactly the right number of cells in a connected sequence.
ANCHOR FROM THE CLUES
The given numbers are your anchors. For each pair of consecutive anchors (say 5 and 9), the path between them must visit exactly the right number of cells — in this case, 3 intermediate cells (6, 7, 8) to get from 5 to 9.
The first question for every gap: is the shortest possible path between the two anchor cells longer than the gap allows? If getting from 5 to 9 requires passing through at least 4 intermediate cells (because the anchors are far apart on the grid) but the gap only has 3 slots, the puzzle has an error — or, more usefully, those anchor placements force the path to take a specific short route.
Sort gaps by size. A gap of 1 between two anchors means the anchors must be adjacent — no intermediate cells. That's immediately forced. Gaps of 2 or 3 have very few routing options on a constrained grid.
DEAD-END DETECTION
A dead-end is a cell that has only one free (unfilled) neighbour. If a dead-end cell is not the start (1) or end (N) of the path, the path must both enter and exit it — which requires at least two free neighbours. A dead-end in the middle of the path is impossible, so the one free neighbour must be either the predecessor or successor of the dead-end cell's number.
This is the most productive technique in Number Mazes: after every placement, scan for cells that have become near-dead-ends (one free neighbour). These force the next number almost automatically.
Corner cells on a square grid start with only two neighbours. As surrounding cells fill, they can become dead-ends with a single remaining free neighbour — check them early.
CONNECTIVITY PRUNING
As the path fills in, check that remaining empty cells are still reachable from both directions of the path. If a cluster of empty cells becomes disconnected from the rest of the grid (surrounded by filled cells on all sides), the path cannot reach them — which means you've made an error somewhere.
On harder puzzles, connectivity checks serve as a pruning tool: if placing a number in a cell would cut off a cluster of empty cells from the remaining path, that placement is forbidden. This rules out choices that look locally valid but break the global path.
HOW IT COMPARES TO HIDATO (STARLINK)
Number Mazes and Hidato (Starlink in GridJoy) are the same concept — fill a grid with a consecutive numbered path — with one key difference: adjacency. Number Mazes use four-directional adjacency (up, down, left, right only). Hidato uses eight (adding the four diagonals).
Four directions means the path is more constrained. Dead-ends appear sooner and corner cells become trapped faster. Hidato's diagonal adjacency gives the path more routing freedom, which makes gap analysis harder but makes corner cells cheaper starting points. On a square grid, Number Mazes fill from the corners outward; Hidato paths thread through them more freely.
THE SOLVE ORDER
- List all gaps between given numbers. Sort smallest first. Fill any gap of 1 (adjacent anchors) immediately.
- For small gaps (2–4), enumerate the possible routes between the anchor pair. If only one route fits the available cells, fill it.
- After each placement, scan for dead-end cells — those with only one remaining free neighbour that isn't a path endpoint.
- Apply connectivity checks: confirm remaining empty cells are still reachable in a connected cluster.
- Move to the next smallest unfilled gap and repeat.
Most beginner and intermediate Number Maze grids are fully solvable with this order. Large grids introduce longer gaps where gap-enumeration becomes impractical — at that point, dead-end cascades and connectivity pruning do more work than gap analysis.