Play Shikaku Online

Start with the primes

The natural instinct in Shikaku is to attack the biggest number on the grid. It feels like the most information. It is almost always the worst place to begin, because a large clue is a flexible clue — and flexibility is exactly what you do not want when you are trying to pin something down.

A clue tells you the area of its rectangle, so the shapes available to it are its factor pairs. A 12 can be drawn six different ways before you even think about where to position it. A 5 can be drawn two: a strip five long, lying flat or standing up. Prime clues are the most constrained numbers on the board, and constrained is what you want first.

The prime rows are shaded. Every one of them has exactly two shapes, however large the number gets — a 13 is no more flexible than a 2. A 12, by contrast, has six: three times as many shapes to weigh up before you even consider where to put it.

A worked opening

Here is a 6×6 puzzle. Nine clues, one unique solution. Before reading on, look at it and decide where you would start.

Take them one at a time.

• The 3 in the top-left corner. A 3 is either a horizontal strip or a vertical one. It must contain its own clue and no other. Running down from the corner is possible on shape alone — but that column already holds the 6 two squares below, and a rectangle may only ever contain one clue. So it runs along the top row instead, and because the corner pins one end, there is only one place it can sit.
• The 3 in the top-right corner. The same argument, mirrored: running down that column would swallow the 5 sitting two squares below it. So it lies along the top row — and it happens to take exactly the three squares the first 3 left behind.
• The 5 on the right-hand edge. This is the one worth dwelling on. A 5 is prime, so it is a strip five squares long, and the grid is only six wide. Lying flat from the right-hand edge it would have to cover almost the whole row, running straight through the 4 and the 6. Standing upright in its own column, there is exactly one position that fits without swallowing another clue.

Three clues placed, eleven of the thirty-six squares accounted for, and not a single guess. Now look at what you would have got from starting with the biggest number instead: the 6 sitting in the middle of the grid has seven possible rectangles. Nothing about it is decidable yet.

Being honest about the middle of the solve: the forcing argument above gets you three rectangles and then stalls. Nothing else on this grid is forced by shape alone. From there you need the second technique.

Work backwards from the empty squares

Most Shikaku advice tells you to think outwards from the clues. The stronger move is often the reverse: pick an empty square and ask which clues could possibly reach it?

Every square must end up inside some rectangle, and that rectangle must contain a clue. So if only one clue on the whole grid can stretch far enough to cover a given square, that square belongs to it — and often that single fact fixes the rectangle’s orientation completely.

Corners are the best hunting ground, because a corner square can only be reached from two directions. Squares along an edge are next.

Here it is doing real work on the same grid. The 4 in the bottom row has three possible shapes and none of them is forced — the first technique has nothing to say about it. But look at the two squares in the bottom-left corner:

Three possibilities down to one, without a single arithmetic step. Notice that this worked on a composite clue, the kind the first technique is weakest on. The two methods cover for each other: factor counting narrows the primes, reach narrows the rest.

Two checks that cost nothing

1. Add the clues up. They must total exactly the number of squares in the grid, because the rectangles tile it completely with no overlaps and no gaps. The puzzle above sums to 36 on a 36-square grid. If you have transcribed a puzzle from a newspaper and the total is wrong, you have copied it wrong — stop solving and check.
2. Watch for squares nothing can reach. If you have placed rectangles in a way that strands a square out of range of every remaining clue, you have gone wrong earlier. This is usually the first sign of a mistake, and it shows up several moves before you run out of options.

Where solvers get stuck

Two failure modes account for most abandoned Shikaku grids.

Placing a plausible rectangle instead of a forced one. A clue with four possible shapes will usually have one that looks tidy, and tidiness is not evidence. If you find yourself thinking “that must be right” rather than “nothing else fits”, you are guessing, and the error will surface twenty squares away where you cannot trace it back.

Forgetting that a rectangle can be one square wide. A 4 is not necessarily a 2×2. It can just as easily be a strip four squares long, and long thin rectangles threading between other clues are exactly what makes the harder grids hard. The solution above contains a 4 drawn as a single column and a 5 drawn as a single column.

Three things you can check about these puzzles

The grids here are generated fresh each time rather than drawn from a fixed set. Three consequences of how that is done are visible on the board in front of you.

The clues sit off-centre. Generate a few puzzles and watch where the numbers land: they cluster at edges and corners of their rectangles far more often than in the middle. That is because each rectangle gets its clue dropped on a randomly chosen square inside it, not a central one. It also means a clue tells you nothing about where its rectangle sits — only how big it is.

Hard grids have more pieces, not bigger ones. Switch between Easy and Hard at the same grid size and count the rectangles. Hard caps how much of the board any single rectangle may occupy, so instead of a few sprawling shapes you get many small ones with more edges touching. Easy also permits up to two single-square clues; medium and hard permit none, because a lone 1 is a free square.

The clue numbers always add up to the grid. Add them yourself on any puzzle here and you will get exactly the number of squares — 25 on a 5×5, 196 on a 14×14. That is not a design choice, it falls out of the rectangles tiling the grid completely.

The step that matters most is invisible, though. Every finished grid is handed to a backtracking solver that counts how many ways it can be completed, and if the answer is anything other than exactly one, the whole attempt is discarded and it starts again. That is why the promise at the top of this page — one solution, never any guessing — is a statement of fact rather than an aspiration. It is checked on every single puzzle before you ever see it.

Shikaku, and its other names

Shikaku (四角) is short for shikaku ni kire, “divide into rectangles”. It was first published by Nikoli, the Japanese puzzle magazine that also brought Sudoku, Kakuro and Slitherlink to a wide audience.

In English it appears as Rectangles, Divide by Squares, Divide by Box and occasionally Boxes. The rules are the same under every name: divide the grid into rectangles, one clue each, area equal to the clue.

It is often confused with Mosaic, where numbers count filled neighbours rather than describing a region, and with Nurikabe, where numbered regions can be any shape at all rather than rectangles. Shikaku’s defining constraint is the rectangle — no L-shapes, no diagonals, no holes.

More logic puzzles

If you like dividing a grid into regions, these use the same instinct: