How to Solve Number Loop Puzzles

How to play on this site

Tap a line between two dots to draw it. Tap it again to mark it with a cross, which means "no line here", and a third tap clears it. If you would rather mark crosses first, the options swap the order. A long press or a right click places a cross straight away, and you can drag across several dots to draw a run of lines in one go; dragging over lines you already drew erases them.

On a keyboard, the arrow keys move a cursor over the dots. Hold Shift with an arrow to draw a line from the cursor, press X and then an arrow to place a cross, and press U or Backspace to undo. Undo and Restart are under the grid as well.

When a number has exactly its count of lines, it fades so you can see what is left. If you draw too many lines round a number it turns red (you can switch these hints off). Easy grids are 5 × 5, medium 7 × 7 and hard 9 × 9. Your time ranks you, and fewer changes break a tie.

Start with zeros and threes

A 0 is the best place to begin: cross all four of its sides. Those crosses reach the neighbouring squares and often decide them too.

A 3 next to a 0 is almost finished at once. The side they share is crossed, so the other three sides of the 3 are lines.

Two 3s side by side share a side that must be a line, and their two outer sides parallel to it are lines too. Try the other choices and you will find a dot with three lines or a dead end.

Two 3s touching at a corner have their outer corners settled: at the far corner of each 3, both sides are lines.

Corners and edges of the grid

The dot in a corner of the grid has only two lines leading from it, so the loop either turns through it or stays away.

• A 1 in a corner can never use the corner sides, because using one means using both. Cross them.
• A 3 in a corner must use both corner sides, because it can leave out only one side and it cannot leave out a corner side alone.
• A 2 in a corner always pushes the loop along the border: the next line along each edge of the grid, just past the 2, is part of the loop.

Along the edges of the grid, the same counting works on the dots that have only three lines.

The dot rule

The rule that does most of the work is about the dots. Every dot has either no lines or exactly two. A loop passes through a dot or misses it, so a dot with one line or three lines is never right.

That gives four quick checks you can run anywhere:

• A dot with two lines is full: cross every other line at that dot.
• A dot with one line and only one open way out must continue that way.
• A dot with one line and all its other ways crossed is a mistake somewhere.
• A dot with every way but one crossed can never be used: cross the last one too.

Put the dot rule next to the numbers and the grid opens up. If the loop runs into a corner of a 3, it has to turn into one of the two sides of the 3 at that corner, so the 3 misses one of them; then the two sides of the 3 away from that corner must both be lines. If the loop runs into a corner of a 1, it uses one of the two sides of the 1 at that corner, so the two sides away from that corner are crosses.

Never close a small loop

The answer is one loop, so you must never join the two ends of a piece unless that join finishes the whole puzzle. When two loose ends sit one line apart, and the rest of the grid still needs lines, cross the line between them.

This rule is easy to forget and very strong in the middle of a hard grid. Look for long pieces whose ends have curled back towards each other: closing them would leave other lines stranded, so the gap between them is a cross, and the ends must find another way.

Inside and outside

A finished loop splits the squares into two groups: inside the loop and outside it. That gives another way to reason.

• Two squares with a line between them are on opposite sides of the loop.
• Two squares with a cross between them are on the same side.
• Everything beyond the edge of the grid counts as outside.

So a square on the border whose outer side is a line is inside, and you can follow crosses and lines from it to name its neighbours. A 3 surrounded by outside squares on three sides is inside, and so on. When you finish a grid on this site, the inside squares fill in, so you can see the shape your loop has drawn.

When you are stuck

Hard grids sometimes need one step of looking ahead. Pick a line near the action, ideally one beside a number or at the end of a piece, and imagine it drawn. Follow the dot rule and the numbers for a few steps. If that ends with a number over its count, a dot with three lines or a closed small loop, the line must be a cross. This is not guessing: it proves the line is wrong, so you can cross it with confidence.

Two more habits help:

• Count the ends. Every loose end must meet another one. If an end has only one open way to go, take it.
• Use crosses freely. Marking where the loop cannot go is half the work, and crosses make the next dot easy to read.

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. Start with easy if you are new: its numbers are close together and the plain rules finish it. Medium and hard show fewer numbers and lean on the dot rule and the small loop rule more.

If you enjoy this kind of reasoning, try Light Up, Bridges and Tents & Trees next.