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Bans piling up in a grid until one square is forced Three copies of the same four by four grid, side by side. In the first, one square is filled in and every other square along its row and down its column carries a small cross, marking where that animal is now banned. In the second a further square is filled lower down, and inside the heavily outlined box in the top right corner three of the four squares now carry a cross, leaving exactly one square clear. In the third that one clear square has been filled in, because nothing else could go there.

When No Row Has Only One Gap

About 12 minutes

The animal grid you solved came apart quickly, and it is worth knowing why. Every row in it was missing exactly one animal. So every row solved itself, and you never once had to look at a column or a box.

That was a puzzle about what is left over. Sudoku is meant to be a puzzle about where things cannot go — and here is a grid that makes you do it.

The grid that will not start

Same four animals. L = Lion · M = Monkey · Z = Zebra · E = Elephant.

A four by four grid where the top right box is being searched for the Lion A four by four animal sudoku divided into four boxes of two by two. L stands for Lion, M for Monkey, Z for Zebra and E for Elephant. Row one reads L, blank, Z, blank. Row two reads blank, E, blank, M. Row three reads M, blank, E, blank. Row four reads blank, Z, blank, L. No row, column or box is missing only one animal. The top right box is drawn with a heavy outline and its four squares are the ones being examined. Z M L E M E Z L
Eight squares are filled and eight are empty. The heavy outline marks the top right box.

Go looking for the move you already have. Take any row and count the gaps: two. Any column: two. Any box: two. There is no row, no column and no box anywhere in this grid that is missing only one animal.

The move you know asks what can go in this square? — and it needs somewhere nearly full to answer from. Nothing here is nearly full. On this grid that question has no answer at all, on any of the eight empty squares.

The grid is not harder. It is asking for a different question.

Turn the question round

Instead of standing on a square and asking what fits, stand on an animal and ask where it is allowed to go.

Take the Lion, and take the top right box — the four squares inside the heavy outline. The Lion has to be somewhere in that box, because every box holds all four animals. So there are only four places it could be, and you can test them one at a time.

A four by four grid where the top right box is being searched for the Lion A four by four animal sudoku divided into four boxes of two by two. L stands for Lion, M for Monkey, Z for Zebra and E for Elephant. Row one reads L, blank, Z, blank. Row two reads blank, E, blank, M. Row three reads M, blank, E, blank. Row four reads blank, Z, blank, L. No row, column or box is missing only one animal. The top right box is drawn with a heavy outline and its four squares are the ones being examined. Z M L E M E Z L

Four squares, and the Lion has to live in one of them. Take each square in turn.

  • Row 1, column 3 — the Zebra is already sitting here. Nothing else can be, so the Lion is out.
  • Row 1, column 4 — empty, but look left along row 1. The Lion is already in this row, over at column 1. One animal per row, so the Lion cannot come back into row 1 anywhere. Out.
  • Row 2, column 4 — the Monkey has this one. Out.
  • Row 2, column 3 — nothing sits here, no Lion is in row 2, and no Lion is in column 3. Three of the four squares are ruled out, so this is the only place left. The Lion goes here, and you never needed to know what else was going in that box.

That is the move. Not what fits here, but where can this go — and the answer arrives when everywhere else has been shut.

A ban travels further than it looks

The interesting part of that is the second square. The Lion that blocked it is at row 1, column 1, which is in a completely different box, three squares away, and it never touches the top right box at all.

It does not have to. It bans the Lion from the whole of row 1, and a box sits across two rows. So one animal, sitting well outside a box, can shut down half that box's squares without being anywhere near it.

That is why the puzzle is worth doing on the boxes rather than the rows. A row ban is a long thin thing that reaches right across the grid. Where it passes through a box, it takes out two squares in one stroke.

Your turn on the same grid

The Monkey is sitting at row 2, column 4. Now look at the top left box — rows 1 and 2, columns 1 and 2 — and work out where the Monkey has to go in it.

In the top left box, which square must the Monkey take?

The Monkey already owns a square in row 2, so both of that box's row 2 squares are shut. Only row 1 is left, and row 1 column 1 is the Lion's. One square remains, so the Monkey takes it — and again you never asked what fits.

When even that stalls

On a full nine-by-nine grid you will meet positions where neither question answers on its own. Two moves get you moving again, and neither is a guess.

Both are the same habit as the Lion: keep shutting doors, and let the last one open by itself. A real sudoku never needs a guess, and now you have the move that makes that true.