Number Link Puzzle: Connect Matching Numbers Without Crossing

What are the rules of number link?

Number link has three rules: matching numbers are joined by a single line, lines never cross or overlap, and when the puzzle is finished no cell is left empty.

Lines move from cell to cell horizontally or vertically only, never diagonally. Only one line may pass through a cell, and a line cannot run over itself. The numbered cells are where lines start and end; a line cannot pass over another number.

The third rule is where the challenge lies. Joining every pair is often easy; doing it while using every cell forces you to think through each route. The game on this page tells you when all pairs are joined but empty cells remain.

Where should you start a puzzle?

The quickest way into a puzzle is through the corners, the edges and the pairs closest together, because those are the places where a line has the fewest possible routes.

A number in a corner has only two neighbouring cells, so its line usually has to run along the edge. When two matching numbers sit near an edge, joining them with a line that follows the edge is the move most likely to be right — and it leaves the middle of the board for the other pairs.

Join pairs that are two or three cells apart early as well. Short routes offer few choices and settle part of the board; as the free area shrinks, where the long lines must go becomes obvious.

Which solving techniques work on hard levels?

Four techniques do most of the work on hard number link puzzles: hugging the edge, finding bottlenecks, avoiding dead cells and drawing the longest line last.

Hugging the edge: once a line touches the edge of the board it usually continues along it. Leaving a one-cell gap between the edge and a line is a mistake unless another line can fill that cell.

Finding bottlenecks: when a one-cell corridor remains between two lines, only one line can pass through it. Working out which pair owns the corridor often unlocks the whole puzzle.

Avoiding dead cells: an empty cell walled in on three sides by lines cannot be filled unless it holds a number. If a line you are drawing leaves such a cell behind, change its route.

Drawing the longest line last: trying to join a pair at opposite ends of the board early blocks everyone else. Place the short, certain lines first; the long line then becomes the filler that takes up the remaining space.

How do board size and difficulty change?

As the board grows, both the number of cells and the number of pairs increase and the possible routes multiply quickly, so difficulty rises with board size.

A 5×5 board has 25 cells and 5 pairs; it takes a few minutes and is ideal for learning the game. 6×6 and 7×7 boards hold 6–7 pairs and call for techniques such as edges and bottlenecks. An 8×8 board has 64 cells and 9 pairs, a 9×9 board 81 cells and 10 pairs; at these sizes a line sent the wrong way often only shows itself a few moves later and has to be undone.

Levels within a size are independent: a higher level number does not make the puzzle harder, only different. To raise the difficulty, move up a board size.

How are the puzzles made, and are they always solvable?

Yes, every puzzle here can be solved, because each one is built backwards from its solution.

First a single long route that passes through every cell of the board exactly once is drawn at random. That route is then cut into pieces of at least three cells, and the two ends of each piece are marked with the same number. This guarantees at least one complete solution. Puzzles with a pair whose ends sit side by side — a pair already solved — are thrown away and drawn again.

A puzzle may have more than one solution. The game checks the rules, not a particular answer: if every pair is joined and every cell is filled, the route you took does not matter.

Where does number link come from?

Number link is a pencil-and-paper puzzle that became widely known through the magazines of the Japanese puzzle publisher Nikoli; it appears as Numberlink in English and Arukone in Japanese.

In its classic paper form letters are sometimes used instead of numbers, and a unique solution is expected. With smartphones, the same rule reached a far larger audience through games that use coloured dots in place of numbers. The game on this page combines both traditions: every pair has a colour and a number, so players who find colours hard to tell apart can play by the numbers alone.

What skills does number link exercise?

Number link mainly exercises spatial reasoning, planning ahead and sustained attention.

Before drawing a line you have to picture whether it will block other pairs a few steps later. When you make a mistake, finding the decision that went wrong and correcting it from there builds patience and systematic problem solving. Because puzzles are short, they also make a good few-minute mental break between classes, on a commute or away from work.

If you enjoy thinking with numbers and shapes, the online abacus in the education section and the maths calculators feed the same curiosity.