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 a 5×5 grid, the numbers sum to 25. On a 10×10 grid, 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).

On mobile, tap-and-drag is tricky on small screens, so there's a tap-two-corners mode: tap the first corner, then tap the opposite corner. Long-press to remove. Clicking a numbered cell without dragging shows ghosted outlines of all valid rectangle placements for that number — a handy way to explore your options.

Worked example: 5×5 grid

Picture a 5×5 grid. 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 — Factorize the 6. The center “6” has options: 1×6, 2×3, 3×2, or 6×1. Some of these extend past the grid boundary or overlap the rectangles you already placed. Usually only one or two factorizations remain valid. If 2×3 is the only shape that fits without bumping into existing rectangles, 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

1. Forced rectangles

Some numbers have only one possible rectangle that fits the grid. A “2” crammed into a corner can only extend in one direction. A “1” is always a single cell. Numbers along edges have fewer options than numbers in the middle. Start by scanning for these — they give you free placements that constrain everything around them.

2. Factorization analysis

For each number, list every way it can form a rectangle. 12 = 1×12, 2×6, 3×4, 4×3, 6×2, or 12×1 — six options. Then check which ones actually fit in the available space. Does the 1×12 extend past the grid edge? Does the 2×6 overlap a rectangle you already placed? Eliminate the impossible ones. If only one remains, place it.

3. Large-number anchoring

Big numbers are your friends. A clue of 15 on a 15×15 grid can only be 1×15, 3×5, 5×3, or 15×1. That's four shapes, and at most one or two fit the available space. Placing a large rectangle carves the grid into smaller sub-regions that constrain neighboring numbers. Tackle these before the ambiguous small numbers in the middle of the grid.

4. Boundary propagation

Every time you place a rectangle, the unclaimed cells around it become more constrained. A neighbor that previously had three valid placements might drop to one. This cascading effect is the main engine of medium and hard solves — one placement triggers the next, which triggers the next. After placing a rectangle, always re-scan its neighbors.

5. Isolated cell detection

If an unclaimed cell can only be reached by one number's rectangle options, it must belong to that number. This comes up when rectangles you've already placed wall off certain cells, leaving them accessible from only one direction. Identifying these cells narrows down the remaining placements.

6. Elimination by contradiction

When you're stuck, pick a number with two remaining options and mentally test each one. If placing a rectangle for number A in position X would make it impossible for a neighboring number B to form any valid rectangle, then position X is wrong for A. This technique breaks open stalled puzzles on expert and einstein grids.

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 5×5 grids are a good starting point for kids learning multiplication tables. Numbers stay small (typically 1 through 6), and the spatial component is manageable. As they get comfortable, 7×7 and 10×10 grids introduce larger numbers with more factorization options. The difficulty scales with mathematical maturity in a way that most puzzle types don't.

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 actually more constrained — they have fewer valid placements. A “15” on a 15×15 grid has at most 4 factorizations, while a “6” in the middle of the grid might have 10+ valid rectangles. Start with the constrained numbers.

Difficulty levels

Shikaku on ThePuzzleLabs comes in five levels that differ by grid size and the techniques needed:

• Easy (5×5): Most numbers have one forced rectangle. Good for learning the rules and getting used to the rectangle-drawing interaction.
• Medium (7×7): Numbers start having multiple factorizations. Elimination between 2–3 candidates becomes necessary.
• Hard (10×10): Boundary propagation is the main technique. Placing one rectangle cascades constraints to its neighbors.
• Expert (12×12): Multi-step constraint chains across 144 cells. Large numbers with many factorizations appear.
• Einstein (15×15): 225 cells of deep factorization ambiguity. Solvable by logic alone, but the chains are long and the interactions are dense.

Frequently asked questions