Shikaku

Shikaku asks you to cut the whole grid into rectangles so that every rectangle contains exactly one number, and that number equals the rectangle’s area — its width times its height. Also published as Divide by Box or simply Rectangles, the puzzle starts from a grid sprinkled with numbers, and squares count as rectangles too.

What does each number tell you?

Two consequences follow from that single idea. Every cell must end up inside exactly one rectangle, with no gaps and no overlaps, and because each number names an area, it also limits the possible shapes. A 6, for instance, can only be a 1×6, 6×1, 2×3 or 3×2 block — and it must be placed so that it covers its own number and no other.

Why every board has one solution

Those constraints are enough to force a unique answer. Each number is a hard area-and-position clue, and the “cover every cell once” rule ties them together. Every board on this page is generated and checked to have exactly one solution reachable by logic alone, so the grid always tiles perfectly without guesswork.

To solve Shikaku, place the forced rectangles first — those at corners, edges and prime numbers — then re-examine the shapes that newly conflict or newly fit. Shikaku is a puzzle of shapes and elbow room, and a few techniques do most of the work:

• List the shapes. For each number, picture the rectangles that fit: a 5 is only 1×5 or 5×1; any prime number has just two shapes, which makes it a great starting point.
• Start at the corners and edges. A number near a corner has fewer ways to extend, so its rectangle is often forced — especially 1s (always a single cell) and the long, thin primes.
• Respect other numbers. A rectangle may contain only its own number, so any shape that would swallow a second number is out. Where two numbers compete for the same cells, usually only one rectangle can legally claim them.
• Fill the forced cells. A cell that only one number’s possible rectangles can reach must belong to that number — claim it and the surrounding shapes tighten.
• Work the leftovers. As the grid fills, an isolated empty cell must connect to the one number whose rectangle can still stretch to it.

Should you ever guess in Shikaku?

No — on a proper Shikaku the next rectangle is always provable. The reliable habit is to place every forced rectangle first, then re-examine the shapes that newly conflict or newly fit. Never guess: the next move usually comes from a corner, a prime number, or a cell only one number can cover.

A corner 2 can only be a 1×2 or 2×1 block extending in one of two directions, so a single blocking neighbour forces it completely. A quick case shows the logic. Imagine a 2 sitting in the very corner of the grid — it is either the corner cell plus the one to its right, or the corner cell plus the one below. If a neighbouring number already blocks one of those directions, the rectangle is fully forced.

How do prime numbers resolve?

Now take a 3 a little further along. Three is prime, so it is a 1×3 or 3×1 strip — never an L-shape or square. If a vertical 3×1 strip would overlap another number, only the horizontal 1×3 remains, and you can draw it in. Each rectangle you commit removes cells from every other number’s options.

Keep claiming the forced shapes and the grid tiles itself: prime numbers and corners give the openings, the “one number per rectangle” rule prunes the rest, and the whole board partitions cleanly — no guessing needed.

What about the bigger numbers?

The bigger numbers simply wait their turn. A 6 seen early might be a 1×6, 2×3, 3×2 or 6×1 block — too many options to place safely — so you leave it while the forced primes and corner shapes carve away its possibilities. By the time you come back, the surrounding rectangles usually leave just one legal way for it to sit. That is the whole art of Shikaku: commit only what is certain, and the flexible numbers resolve themselves as the grid closes in.

This page offers Shikaku at 5×5 and 7×7, with difficulty driven mainly by how much freedom the numbers have. The smaller grid is a relaxed introduction, with numbers close enough that most rectangles are quickly forced; the larger grid leaves more open space, so a single number can have several plausible shapes until its neighbours rule them out.

Difficulty comes mainly from how much freedom the numbers have. Tightly packed grids with many small numbers solve fast; sparser grids with bigger numbers — which have more possible rectangles — ask for more careful elimination. Tap New for a fresh layout at either size, or play the shared Daily for the same grid as everyone else that day.

Shikaku comes from the Japanese instruction to “divide into rectangles”, and in English it is also sold as Divide by Box or Rectangles — the same puzzle under different labels.

It is a region-division puzzle — you carve the grid into pieces under a rule — which makes it a cousin of the shading-and-region puzzles here. If you enjoy partitioning a grid, try Nurikabe, where numbers define islands within a connected sea, or Nonogram, where numbers describe runs of shaded cells. All three turn a handful of numbers into a single, exact layout.

The key Shikaku terms are clue (a number naming an area), rectangle (the block containing one clue), area (width × height), and tiling (the finished gap-free partition). A few terms recur in Shikaku guides:

• Clue (number) — a digit that names the area of the rectangle it belongs to.
• Rectangle — an a×b block of cells (squares included) containing exactly one clue.
• Area — width × height; it must equal the clue inside.
• Tiling — the finished partition, where every cell sits in exactly one rectangle with no overlaps.

You don’t need the words to play — just drag a rectangle — but they make strategy guides clearer.

The most common Shikaku mistakes are guessing a shape, putting two numbers in one box, leaving gaps, and ignoring primes and corners. Most Shikaku slips come from a few habits:

• Guessing a shape. Drawing a plausible rectangle you cannot prove can leave an orphan cell that no number can reach later. Find the forced shape instead.
• Two numbers in one box. A rectangle may contain only one clue; it is easy to stretch a shape over a second number by accident.
• Leaving gaps. Every cell must belong to a rectangle — an empty cell at the end means something earlier was wrong.
• Ignoring primes and corners. These are the easiest forced moves; skipping them makes the puzzle feel harder than it is.

Shikaku is quietly a puzzle about factor pairs: a number’s factors are its possible rectangle shapes, which is the single most useful trick in the game. Every number is an area, and an area can only be made by rectangles whose side lengths multiply to it — so listing a number’s factors instantly lists its possible shapes.

• A prime (2, 3, 5, 7, 11 …) has only one factor pair, so it is always a single straight strip — the easiest kind of shape to place, and a great place to start.
• A perfect square (4, 9, 16) can be a neat square or a long strip, giving a little more freedom.
• A composite like 12 (1×12, 2×6, 3×4 and their rotations) is the most flexible, so it is usually best left until its neighbours have trimmed the options.

How does factor-reading change the grid?

Reading the numbers this way changes how the grid feels. Instead of staring at open space, you see each clue as a short menu of rectangles, then use position, the edges and the no-overlap rule to pick the only one that fits. It is the same factor-and-fit reasoning whether the board is 5×5 or 7×7 — the larger grid simply offers each number a little more room before its neighbours close in.

Shikaku is calm, visual and entirely logical — the pleasure of carving a messy grid into tidy, interlocking pieces until everything fits with no gaps and no overlaps. There is no arithmetic beyond reading an area, no luck, and a single answer always waiting to be deduced. A quick 5×5 is a relaxing few minutes; a 7×7 is a more involved spatial workout for when you have time to think.

Because it is fair by design, progress always feels earned, and the shared Daily gives you the same grid as everyone else to compare against. Shikaku also trains a distinct skill from the number puzzles — partitioning space rather than placing digits — which makes it a refreshing companion to the shading and region puzzles in this collection. If you like turning numbers into a single exact layout, try Nurikabe and Nonogram next.

Shikaku is a puzzle from the Japanese publisher Nikoli, the house that refined and named so many of the pencil puzzles in this collection. Its name comes from the Japanese phrase for dividing the grid into rectangles.

Like its stablemates Sudoku and Slitherlink, Shikaku spread internationally and picked up plain-English names — Divide by Box and Rectangles — while keeping the same elegant premise: a grid of numbers that, read as areas, partitions the board into one and only one set of rectangles, all by logic.