Hitori

The patterns that shade themselves

Hitori asks you to black out repeated numbers until none shows twice unshaded in any row or column, and the trick to starting fast is recognising a few shapes that resolve on sight. Three of the same number in a line is the classic one: shade the two outer cells and leave the middle white. It cannot go any other way. Shading only the centre would still leave the two ends as a matching white pair, and shading both ends is the single arrangement that keeps one copy white without placing two black cells side by side.

The sandwich is just as quick. When the same number sits two cells apart in a line, with a single different cell between them, that middle cell has to stay white. Here is the proof. Shade the middle cell and both of its neighbours are forced white, because shaded cells never touch. But those two neighbours are the matching pair, and two identical white cells in one line is exactly what the puzzle forbids. So the middle cell stays white, every time.

One more freebie: when two identical numbers sit directly beside each other, exactly one of them ends up shaded. They cannot both stay white, since that is a repeat, and they cannot both be black, since black cells never touch. You may not know which one goes yet, but any third copy of that number in the same line must be shaded, because one of the pair is guaranteed to stay white. Spotting these three shapes before anything else routinely settles a tenth of the grid in the first minute.

Two rules keep it honest

Shading is not free. Two black cells may never touch edge to edge, so the instant you shade a cell, all four of its orthogonal neighbours are forced white, which often resolves the very duplicate you were staring at. And every white cell must stay connected to the rest as one continuous group, so you cannot shade a cell if doing so would split the white region in two or cut off a corner. Corners are the usual crime scene here: a corner cell has only two neighbours, so it takes just two black cells to smother it.

• If a cell is boxed in by shaded neighbours, or shading it would isolate a patch of white, it has to stay white.
• Once you shade one cell of a duplicate pair, its partner is automatically the safe white one.
• Numbers that appear only once in their row and column are the backbone of your connected white path.
• Watch the borders. Two shaded cells on the diagonal beside a corner wall that corner off, so once one of them goes black, the other must stay white.

A routine for when you stall

Stuck grids almost always crack one of three ways. First, rescan for the patterns above, because every new shade creates fresh sandwiches and pairs nearby. Second, pick the number that repeats most in a single row and check its columns too; a cell that duplicates in both directions at once is nearly always the one to shade, since one move fixes two problems. Third, run a quick test in your head: suppose a doubtful cell stays white, follow the shades that forces, and watch for a contradiction such as a stranded white cell or two blacks touching. If the contradiction comes, the cell must be shaded. That is not guessing; it is proof by elimination, and small grids rarely need more than one step of it.

Alone, by name

Hitori comes from Nikoli, the Japanese publisher behind the modern Sudoku boom, and first appeared in its magazine in 1990. The full name is Hitori ni shite kure, roughly leave me alone, a good fit for a puzzle about leaving exactly one of each number standing per line. It never caught on the way Sudoku did, which is a pity. Sudoku is about filling empty cells; Hitori hands you a full grid and asks you to take away. The interplay between the shading rule, the no-touching rule and the connection rule makes solving feel like a short logic proof, and the fastest solvers are the ones who trust those rules enough to let the grid shade itself.