How to solve Futoshiki

Every row and every column holds the digits from 1 up to the number of its cells, each one once — on a 5 × 5 that is 1 to 5.

A sign between two cells points at the smaller of the two digits: 2 < 5, and 5 > 2. Between a cell and the one below it, it reads the same way.

A few digits come with the board; the signs carry the rest.

Tap a cell and pick a digit on the pad under the board; the same digit again clears it. With notes on, digits are pencilled small in the cell instead. The digits the board came with stay put, and the signs take no taps — a tap on one lands on the cell under your finger.

A reddish cell holds a digit its row or column already has, or a digit on the wrong side of a sign — and that sign reddens with it.

A board comes with a few digits and a scatter of signs; the rest you fill in yourself. Every board has exactly one solution, and it can be reached without guessing. A 5 × 5 needs only the first three moves on this page, a 6 × 6 the whole-line read as well, and a 7 × 7 the two-line read too.

A sign points at the cell holding the smaller of the two digits. That is a floor and a ceiling in one: the cell on the open side cannot be the 1, and the cell the point faces cannot be the largest digit. Put that together with the digits the line already holds and a cell is often settled outright.

Signs link up. Where three cells rise one after another, the first has to leave room for two digits bigger than itself and so cannot be large, and the last for two smaller and so cannot be small — and a whole line of rising signs is simply the digits in order. The longer the chain, the more it says.

The question can be turned round: not “what can stand in this cell” but “where in this row can this digit stand”. Every digit has exactly one place in every line, so when the signs and the other lines rule it out everywhere but one cell, that is where it goes.

Sometimes no single cell can be settled, and the line has to be read whole: which digits are still to be placed in it, and which ways of laying them out its own signs allow. A digit that no surviving arrangement puts in a cell can be struck off that cell, even when nothing has been placed.

Two lines can be held against each other. If a digit’s only places in two rows lie in the same two columns, then those two columns take that digit in those two rows — whichever way round — so it stands nowhere else in them. The hardest boards turn on exactly this.