How to solve Suguru

The board is cut into regions of one to five cells. Fill every region with the digits from 1 up to the number of its cells, each once — a region of three holds 1, 2 and 3, a region of five the digits 1 to 5.

Two cells that touch — along a side or at a corner alone — never hold the same digit, whether or not they share a region.

Some digits come with the board.

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.

A reddish cell holds a digit its region already has or has no room for, or a digit touching its like.

The digits given at the start stay; the rest you fill in yourself. Every board has exactly one solution, and it can be reached without guessing. A 6 × 6 needs only the first three moves on this page, an 8 × 8 the shared neighbour as well, and a 10 × 10 the pair too.

Every cell has up to eight neighbours and must differ from each of them. On top of that comes its own region, which says which digits are in play at all. When all of that leaves one digit, the cell is settled.

A region holds each of its digits exactly once. When only one of its cells is still empty, there is no need to look at the neighbours — just check which digit the region is still missing. A region of one cell is simply a 1.

The question can be turned round: not “what can stand in this cell” but “where in this region can this digit stand”. A region must hold each of its digits, so if the neighbours rule a digit out of every cell but one, that is where it stands.

Sometimes it is not yet known which cell of a region a digit will take, only that it is one of two or three. That is already enough: a cell outside the region that touches every one of those places cannot hold the digit — wherever it lands, it will touch that cell.

Two cells that see each other — they touch, or lie in one region — must hold different digits. If both are down to the same two possibilities, they will share them out between themselves. Which takes which is not yet known, but both digits are spoken for, so a cell that sees both can hold neither.