How to Solve Number Path Puzzles

How to play on this site

Select a square by tapping it or moving to it with the arrow keys, then type a number or tap it on the number pad. Two-digit numbers are typed one digit after the other, quickly, and count as one entry. Erase clears a square, and Undo steps back one entry at a time.

Next number mode is the fast way to play. The page keeps track of the lowest number still missing after the path you have built, and a single tap on a square places it there; the next tap places the one after. The squares where that number could still go are lit in green (you can turn this off in the options), and the square holding the end of your current run has an orange ring.

Notes mode writes small candidates in a square instead of a full number, which is handy when two numbers are fighting over the same place. With mistakes shown, a number used twice or two consecutive numbers that don't touch turn red. The thin line across the board joins every pair of consecutive numbers you have placed, so you can see the path grow.

Start with the gaps

The givens split the path into short runs. Between 7 and 10, for example, two numbers are missing: 8 and 9. Look at every gap and sort them by length. A gap of one number is often decided at once. If 7 and 9 are given, 8 must touch both of them, and there are rarely more than two squares that do. If only one of those squares is still empty, 8 goes there.

A gap of two works the same way with one more step. 8 must touch 7, 9 must touch 10, and 8 and 9 must touch each other. Write out the squares next to 7 and the squares next to 10, and look for a touching pair, one from each list. Small gaps near the edge of the grid have very few choices.

Work through every short gap before you look at the long ones. Each number you place takes a square away from the others, and that is what unlocks the long gaps later.

Count the distance

The most useful idea in the puzzle is distance. Because a step can go diagonally, the distance between two squares is the larger of the row difference and the column difference: a square two rows down and five columns across is five steps away. That is the fewest steps any path can take between them.

So if 3 and 8 are given, they are five numbers apart and can be at most five squares apart. Turn it round, and you can rule out squares. Number 5 is two steps from 3 and three steps from 8, so it has to sit in the overlap of a box two squares round 3 and a box three squares round 8. On a 7 × 7 grid that overlap is often only a handful of squares.

When two givens are exactly as far apart as their gap allows, the path between them has no room to wander. If 20 and 24 are four columns apart in the same row, 21, 22 and 23 march straight along that row, one column at a time, or step up and down diagonally while still moving one column each time. Tight gaps like this settle quickly.

Corners, edges and one-way squares

Every square in the middle of the path is entered from one number and left for the next, so it needs two neighbours on the path. A corner square has only three neighbours, and an edge square five. When most of a square's neighbours are already taken by numbers that can't link to it, the square is in trouble.

The rule to look for: an empty square with only two empty or useful neighbours left must be passed through using those two. If an empty corner has one empty neighbour and one neighbour holding 14, then whichever number goes in the corner must link to 14 and to that empty square. So the corner holds 13 or 15, and the empty square holds 12 or 16.

An empty square with only one useful neighbour is even stronger: it can only be the very start or end of the path, and since 1 and the last number are given, that means something has gone wrong earlier. Check your entries.

Squares only one number can reach

Sometimes the trick is to look from the square's side instead of the number's side. Pick an empty square and ask which missing numbers could still live there. Count the distance to the nearest givens. If the square is far from everything except the gap between 30 and 33, then 31 or 32 must take it, because every open square has to hold some number. This is the same reasoning as a hidden single in sudoku.

It helps most late in the puzzle, when a few lonely squares are left in a corner and only one gap is close enough to reach them.

Blocked squares

Some medium and hard grids have two or four shaded squares, always placed in pairs that mirror each other through the centre. They never hold a number and the path can't step onto them, but it can step diagonally past them. A blocked square makes its neighbours harder to reach, which often turns them into one-way squares near the edge. The total count also changes: on an 8 × 8 grid with four blocked squares the numbers run from 1 to 60, not 64.

A worked habit

A good routine for every grid:

1. Read off the gaps between consecutive givens and note the short ones.
2. Fill every gap of one or two numbers that has a single answer.
3. Check corners and edges for squares with only two ways out.
4. For each longer gap, use distance to shade where its middle numbers can't go.
5. Look at the remaining empty squares and ask which gap can still reach each one.
6. Repeat. Each placement shortens a gap or removes an option, so the steps feed each other.

When a gap has two possible routes, write notes instead of guessing. The route that strands a square somewhere is the wrong one.

Daily play

There is a new grid at each difficulty every day, the same for everyone, and your first solve of the day is ranked by time, with fewer entries breaking ties. Easy grids are 5 × 5 with plenty of givens. Medium grids are 7 × 7 and hard grids 8 × 8, with fewer givens and sometimes blocked squares, so distance and corner reasoning matter more.

If you like this kind of counting puzzle, try Futoshiki, Kakuro and Sudoku next.