Numberlink

Filling every cell is the real rule

Numberlink looks like a simple connect-the-dots, and if joining the pairs were all that mattered it would be trivial. The constraint that makes it a puzzle is the one people underrate: every single cell has to be covered by a path. You cannot just find any route between two matching numbers. You need the set of routes that leaves no empty square behind and never lets two paths cross. That is why a lazy, wandering shortcut between a pair is almost always wrong; it strands cells in the corners that no other path can legally reach. Speed matters too: there is exactly one solution, so the clock is really measuring how quickly you can string deductions together, not how fast you can scribble.

Start at the edges and corners

The corners are where to begin. A number sitting in a corner has only two ways out, and often only one that does not immediately dead-end, so its first step is usually forced. Edges behave the same way: a path running along a wall has fewer choices than one loose in the middle. Work those forced moves first and the interior fills in around them.

Here is a small worked case. Say a 3 sits in the top-left corner and the cell below it is already claimed by another pair's route. The corner 3 now has exactly one exit, to the right, so you can draw that step with total confidence. The cell it just entered becomes easier to read as well, because one of its sides is now spoken for. Forced moves breed more forced moves, and good solvers ride these chains as far as they will go before they even glance at the middle of the grid.

A few checks keep you honest along the way:

• A path should never run alongside itself. In a puzzle with one solution, a route that hugs its own earlier line could be shortened, so if yours does, be suspicious.
• If a cell can only be reached by one pair's path, that path has to go through it. These forced cells chain together into whole routes.
• When a route would wall off an empty pocket that no remaining pair can enter, it is wrong; back it out before you build anything on top of it.

Count the pockets

The strongest habit for hard grids is region counting. Every long path you draw slices the board into pockets, and each pocket has to be filled by the numbers inside it or the paths that pass through it. So pause and count. If a pocket holds a 4 but its partner sits outside, your new line has cut the 4 off from its twin, and something has to change. If a pocket holds no numbers at all and no unfinished path pokes into it, nothing can ever fill it. Spotting a dead pocket three moves early saves you the slow discovery that the last corner cell is unreachable.

The border deserves the same attention. The outside ring of cells is the hardest area to cover, and in most well-made grids at least one pair runs a long stretch along the wall to mop it up. When a far edge looks unreachable, ask which pair could sweep the border on its way across, and test that route first.

The puzzle behind Flow Free

If this feels familiar from a phone screen, that is because Numberlink is the pen-and-paper puzzle the hugely popular app Flow Free is built on. Flow Free arrived in 2012 from a small studio called Big Duck Games and has been installed more than a hundred million times, which makes this quiet magazine puzzle one of the most played logic games ever. The puzzle itself is far older. Historians trace the idea back to Sam Loyd, the great American puzzle maker, who printed an early version in a newspaper column in 1897. In Japan it was taken up by Nikoli, the publisher that made Sudoku famous, and there the rules settled into today's form: one solution, no crossings, fill everything.

One more thing earns Numberlink respect. Computer scientists have proved that solving large Numberlink grids is NP-complete, which puts it in one of the hardest problem classes in computing. The friendly grid you solve over lunch is the small, tame member of a genuinely deep family.