Number path puzzle strategy: 8 techniques with worked boards

Technique 1 · basic

Corners

Every tile the path passes through needs two links: one way in and one way out. A corner tile has only two neighbors, so the path has to use both. That is the first thing to fill in on any board, before you even look at the numbers: draw a short bend in each corner. There is one exception. If the corner holds the 1 or the last number, the path ends there and needs only one link, so the corner tells you nothing yet. Check the corner for a number before you draw.

A bend in the corner also hands you a fact about the next tiles. The tile beside the corner is now reached by the path, and if it has only one other open side, the path goes there too. One corner often runs two or three tiles along the edge.

Where to look: the four corners, on every new board, and check each for the 1 or the last number.

Technique 2 · basic

Dead ends and walls

A wall works like the edge of the board. A tile with a wall on one side and the board edge on another has two sides left, so it behaves like a corner. A tile between two walls is a corridor: the path must go straight through. The same goes for a tile closed in by path you have already drawn: once a neighbor is full, that side is gone. Count each tile's open sides. Two open sides means both are path. A tile that needs two links but has only one open side is a dead end, so one of the links you drew must be wrong.

Walls are where puzzle makers hide the start of a solve. On a board with two or three walls, look at the tiles touching them right after the corners. A wall rarely sits there for decoration: it usually turns one tile into a corridor and starts a run of forced moves.

Where to look: tiles touching a wall, tiles touching two walls, and tiles boxed in by path you have drawn.

Technique 3 · basic

Forced moves along the border

Once the path reaches a tile, that tile needs exactly one more link. Count its open sides again: the side it came in by is used, and the walls, the board edge and full neighbors are out. If one side is left, the path goes through it. Along the border this happens all the time, because an edge tile starts with only three sides. Follow the path from each corner bend and keep extending it while each new tile has a single way on. On easy boards this one habit fills most of the edge.

Stop as soon as a tile has two ways on. Don't pick one by feel. Mark where you stopped and look elsewhere: another corner, a wall or a number usually decides the open choice a few moves later, and then you come back and keep following the edge.

Where to look: the end of every piece of path you have drawn, especially on the edge.

Technique 4 · basic

Work backward from the last number

The path has two ends: the 1 and the highest number. Each end needs only one link, so the end tiles follow their own rule: when only one side is open, the path comes in through it. Most players start at the 1 and draw forward, but the last number is just as strong a starting point and often a better one, because it tends to sit in a corner or against a wall. Treat it as a second start and draw backward toward the number before it.

Then do the same for the stretch between the last two numbers. The path from the 5 to the 6 can't touch any other number, so it has to fit between them. Working from both ends of the path halves the open ground in the middle.

Where to look: the highest number, then the 1. Count their open sides first.

Technique 5 · intermediate

No loops

After a few moves the board holds several pieces of path that don't touch yet. Every piece will become part of the one path, so its two ends must never be joined to each other: that would close a loop, and a loop can't be part of a path through every tile. So when the two ends of one piece sit side by side, rule out the link between them with a cross. Small loops are the ones to watch: a U shape around three sides of a 2x2 square is the classic case.

A cross between two tiles is often more useful than the link itself. Each end tile loses an open side, and if it then has only one way on, the path goes there. Look for loops, cross them out, then count sides again.

Where to look: two ends of the same piece that sit next to each other, most often around a 2x2 square.

Technique 6 · intermediate

Segment by segment: numbers in order

Between two numbers in a row, say the 6 and the 7, the path meets no other number. Think of the solve as segments: 1 to 2, 2 to 3, and so on. A piece of path that already holds the 5 can't link straight to the 7, because it would skip the 6. The same goes for joining two pieces whose numbers don't follow on. Whenever two pieces or a piece and a number sit side by side, read the numbers across the link they would make. If they don't count up by one, cross the link out.

This is the technique that uses the numbers most, so it grows stronger as the board fills. A number with two pieces arriving from different sides is a good sign: those pieces must be the segments before and after it.

Where to look: a piece of path next to a number that isn't the one before or after the number it holds.

Technique 7 · intermediate

Don't cut off a region

Every tile has to be reached, so the path must never seal off a part of the board. Look at a pocket of tiles: a group with few ways in. If it holds neither end of the path, the path has to come in and go out again, so it needs at least two open doors. A pocket with exactly two doors must use both, and any move that would close one of them is wrong. Play it out in your head: a link that fills a door tile from the outside shuts that door for good.

Pockets form along walls, between pieces of drawn path, and in corners the path has run past. When you draw a long stretch of path, glance at the tiles it leaves behind: can the path still get in and out of them?

Where to look: groups of tiles between walls and drawn path with only two ways in.

Technique 8 · advanced

Checkerboard parity

Color the board like a checkerboard. Every step of the path moves to the other color, so the path alternates dark, light, dark, light. A stretch that covers a block with as many dark tiles as light ones must start on one color and end on the other. A block with one more dark tile must start and end on dark. Use it on a pocket with three open doors and no end inside: the path passes through once, using two doors, and the colors tell you which pairs of doors can be its way in and out.

This is the advanced one: reach for it when the basic moves stall. On a 5x5 board there are 13 dark tiles and 12 light ones, so the path starts and ends on dark. On 6x6 there are 18 of each, so its two ends sit on different colors. Hidato solvers use the same count to check a stretch between two numbers.

Where to look: a block of tiles with three ways in, once corners, walls and the numbers have nothing more to give.

Common mistakes

• Drawing a long path from the 1 by feel. It feels like progress, but a guess ten tiles back is hard to find. Draw only links you can prove, and use the crosses as much as the lines.
• Treating a numbered corner as a bend. A corner with the 1 or the last number needs only one link. Check the number before you draw both sides.
• Forgetting the crosses. A link you have ruled out is information too. It takes a side away from two tiles, and that is how most forced moves appear.
• Leaving a pocket behind. A run along the edge can trap a few tiles behind it. Before you extend a piece, check that the tiles beside it still have two ways in.