How to solve Shikaku

What is Shikaku?

Shikaku is a rectangle-partition puzzle from Nikoli, invented in 1989 by Yoshinao Anpuku, a university student in Kyoto. The name comes from the Japanese 四角に切れ (shikaku ni kire), meaning “cut into rectangles.” You might also know it as Cellblocks (the name The Guardian newspaper uses), Divide by Box (Nikoli's official English name), or simply Rectangles.

The premise is straightforward. You get an N×N grid with a few numbered cells scattered across it. Everything else is blank. Divide the entire grid into non-overlapping rectangles so that each rectangle contains exactly one number, and that number equals the rectangle's area. A cell showing “6” must belong to a rectangle covering 6 cells — could be 2×3, could be 1×6, could be 3×2 or 6×1. The grid, the numbers, and the constraint that every cell belongs to some rectangle — that's all you need.

The two rules

Shikaku has the shortest ruleset of any Nikoli puzzle on this site. Most grid puzzles need four to six rules. Shikaku needs two:

1. Divide the grid into rectangles. Every cell must belong to exactly one rectangle. Rectangles cannot overlap. Squares count as rectangles (a 3×3 block is perfectly valid).
2. Each rectangle contains one number equal to its area. No rectangle can be empty. No rectangle can contain two numbers. The number inside tells you how many cells the rectangle covers.

A consequence worth noting: since every cell belongs to a rectangle and every rectangle has exactly one number, the total of all numbers on the grid equals the total number of cells (N×N). On the 6×6 easy boards the numbers sum to 36. On a 10×10 they sum to 100. This is a useful sanity check when solving.

How to play on ThePuzzleLabs

On desktop, click on one corner of where you want your rectangle, drag to the opposite corner, and release. The rectangle fills in with a pastel color and the cells get claimed. To remove a rectangle, right-click on any cell inside it (or press Ctrl+Z to undo).

Dragging works the same way on a phone: press a corner and slide to the opposite one. A single tap is not a half-finished rectangle — it drops a single square there and then, so dragging is the only way to take more than one cell, and a stray tap can never leave a selection armed behind you. Tap a rectangle you have drawn to remove it. Press and hold a numbered cell to see ghosted outlines of every rectangle that number could still take — a handy way to explore your options.

Worked example: a small grid

This example uses a 5×5 grid — smaller than any of the five levels, so the whole solve fits in one read. Picture it: in the top-left corner sits a “3,” the center has a “6,” and the bottom-right area has a “4.” Several smaller numbers (1s and 2s) fill out the rest. The total of all numbers is 25, which is 5×5 — checks out.

Step 1 — Place the 1s. Any cell numbered 1 is its own rectangle: a 1×1 block. Place those immediately. They're free information.

Step 2 — Corner and edge numbers. The “3” in the top-left corner can only extend right (1×3) or down (3×1). If the cells to the right are blocked by another number, it has to go down. Check the constraints and place the only valid option.

Step 3 — Work out the 6. A 6 can be 1×6, 2×3, 3×2, or 6×1, but a 1×6 strip needs six cells in a line and this grid is only five wide, so the 6 has to be 2×3 or 3×2. Check both against the rectangles already placed. If 2×3 is the only one that fits without bumping into them, place it.

Step 4 — Fill in the rest. With the big numbers placed, the remaining unclaimed cells form small regions. The “4” near the bottom-right might be forced into a 2×2 square. The “2s” fill in the gaps. When every cell is claimed and every rectangle has exactly one number matching its area, you're done.

Solving techniques

Four techniques solve every Shikaku board on this site, and they come in order of depth: the first finishes an easy board on its own, and each level above adds the next. A fifth is listed after them — a fallback for when you cannot see the next step.

Before any of them, get into the habit of listing a clue's shapes. A 12 can be 2×6, 3×4, 4×3 or 6×2, and on a board ten wide that is the lot — a 1×12 strip will not fit. Then ask where each of those shapes could sit so that it still covers the clue. That set of placements is what the techniques below whittle down.

1. A clue with only one legal rectangle

Sometimes a clue has exactly one placement left, and you can simply draw it. A “1” is always a single cell. A “2” crammed into a corner can only extend one way. A prime clue can only ever be a single line of cells — one row or one column long — so a 5 pinned against the edges often has nowhere else to go. Scan for these first: an easy board is built out of nothing else, and every harder board still starts here and returns here after each placement.

2. A cell only one clue can reach

When the forced clues run out, stop looking at clues and look at an empty cell instead. Ask which clues still have a placement that covers it. If only one does, that clue must cover that cell — so every one of its placements that misses the cell is out, whichever shape turns out to be right. This is the move medium introduces, and hard asks for it across roughly a third of the solve.

3. A cell every placement of a clue would cover

Turn technique 2 around. Take a clue with several placements left and look for a cell that all of them cover. That cell belongs to this clue already, before you know which placement is right — so every other clue that wanted it can drop the placements that reach it. Expert boards are the first that need this.

4. A placement that strands a cell

A placement can be wrong because of what it leaves behind rather than anything about the rectangle itself. Draw it in your head, then look at the cells it has just walled off: if one of them now has no clue that could ever reach it, the placement is impossible and you can rule it out. Nothing about the rectangle broke a rule — the stranded cell is the proof. This is the deepest step the levels ask for, and einstein is where it shows up.

5. A trial placement refuted by following it through

The last resort, and none of the five levels require it. Take a clue with two placements left, commit to one, and run techniques 1 to 4 until nothing more follows. If that ends in a contradiction — a clue with no placement left, or a cell no clue can cover — the placement you committed to was wrong, so take the other. That is a proof, not a guess: you are not hoping the branch works out, you are keeping only the one that cannot be refuted.

Shikaku for math learning

Nikoli explicitly positions Shikaku as educational: “a great math trainer for kids — while playing the game they learn multiplication.” The reason is mechanical. Every clue number is a factoring problem. When a child sees 12 on the grid, they work through 1×12, 2×6, 3×4 — and then figure out which one fits the physical space. That's area calculation, factoring, and spatial reasoning happening together, and none of it feels like a worksheet.

The easy 6×6 grids are a good starting point for kids learning multiplication tables. Nine or so clues share 36 cells, so the rectangles average four cells each, and every one of them is forced outright by its own number — there is no reasoning to get stuck on, just the arithmetic. The 8×8 and 10×10 levels above have more cells and bigger clues to factor, but what actually makes them harder is the reasoning they ask for, not the size of the numbers.

Shikaku vs other puzzles

Sudoku is the obvious comparison point. Both are grid puzzles from Nikoli, both use numbers as clues, and both require pure logic. But Sudoku is about placing digits so that rows, columns, and boxes have no repeats. Shikaku is about drawing territorial boundaries. The mental model is completely different — Sudoku feels like bookkeeping, Shikaku feels like mapmaking.

Suguru (Tectonic) is a closer relative. Both divide a grid into regions that must satisfy numeric constraints. The difference is that Suguru gives you the regions and asks you to fill in numbers, while Shikaku gives you the numbers and asks you to draw the regions. Inverted problems, similar spatial reasoning.

KenKen shares the arithmetic angle. Both puzzles make you think about factors and products. In KenKen the arithmetic is explicit (cage targets with operations); in Shikaku it's implicit (every number is an area, which is a product of width × height).

Common mistakes

Forgetting that squares are rectangles. A cell numbered 4 can belong to a 2×2 square. A cell numbered 9 can belong to a 3×3 square. People sometimes skip the square factorization when listing options and miss valid placements.

Leaving gaps. Every cell must belong to a rectangle. If you place all the numbered rectangles and unclaimed cells remain, something is wrong. The numbers should sum to N×N, and the rectangles should tile the grid perfectly.

Ignoring large numbers. It's tempting to start with small numbers because they feel simpler. But large numbers are usually more constrained — they have fewer valid placements. On a board ten wide a “15” can only be 3×5 or 5×3, because a 1×15 strip will not fit; a “6” sitting well inside the board has four shapes and a choice of positions for each. Start with the constrained numbers.

Difficulty levels

Shikaku on ThePuzzleLabs comes in five levels, and each step changes exactly one thing. The board grows at medium and expert; the thinking deepens at hard and einstein. So hard is medium's board asking a harder question, and einstein is expert's board asking a harder one.

• Easy (6×6): Around ten clues, and every rectangle is forced outright by its own number. Technique 1 and nothing else — you read each rectangle off its clue and draw it.
• Medium (8×8): A bigger board, and the forced clues no longer finish it. Some rectangles give way only once you spot a cell that just one clue can reach.
• Hard (8×8): Medium's board, asking that same question far more often — about a third of the solve rather than a sixth. No new technique, much more of the one you just learned.
• Expert (10×10): A bigger board and hard's reasoning, plus clues whose every remaining placement covers one shared cell — so that cell is theirs before you know which placement is right.
• Einstein (10×10): Expert's board and the deepest step: ruling a placement out because it would strand a cell that no clue could reach.

Frequently asked questions