Shikaku

Shikaku (also called Rectangles) is a logic puzzle where the player must divide a grid into non-overlapping rectangles (or squares). Some cells contain numbers, and each number indicates the area (in cells) of the rectangle that contains it. Every cell must belong to exactly one rectangle, and each rectangle must contain exactly one number. The challenge is to find the unique way to partition the entire grid into rectangles that satisfy all the size constraints.

Rules

1. Divide the entire grid into rectangles and/or squares.
2. Each rectangle must contain exactly one number.
3. That number indicates the total area of the rectangle (number of cells).
4. Rectangles may not overlap.
5. Every cell in the grid must be covered by exactly one rectangle.

Example

A cell containing "6" must be part of a rectangle with an area of 6 cells. That rectangle could be 1x6, 6x1, 2x3, or 3x2.

Strategies

• Prime Numbers: A prime-numbered cell (2, 3, 5, 7, 11, 13...) can only form a 1xN or Nx1 rectangle, since primes have no other factor pair. This severely limits possibilities.
• Number 1: A cell with "1" is a 1x1 rectangle all by itself.
• Corner and Edge Cells: Numbered cells near corners or edges have fewer possible rectangle orientations, since the rectangle cannot extend beyond the grid.
• Blocking: If placing a rectangle in one orientation would make it impossible to cover a neighboring numbered cell, that orientation is eliminated.
• Gap Filling: Unnumbered regions must eventually be claimed by a neighboring numbered cell's rectangle. If only one numbered cell can reach an empty area, its rectangle must extend there.
• Factorization: For each number, list all possible width × height combinations. Overlay each on the grid considering borders and other rectangles to narrow the choices.

Shikaku was created by the Japanese puzzle publisher Nikoli, one of the world's most prolific puzzle design companies. The puzzle first appeared in Nikoli's puzzle magazine *Puzzle Communication Nikoli* in 2001. The name "Shikaku" comes from the Japanese word for "rectangle" (四角).

Nikoli published the puzzle under the full name "Shikaku ni Kire" (Divide by Squares), though it is commonly abbreviated to "Shikaku" internationally. Like many Nikoli puzzles, it follows the company's design philosophy of simple rules that produce rich logical challenges. The constraint of partitioning every cell into exactly one numbered rectangle creates puzzles that require both local deduction and global awareness.

Shikaku gained international popularity through the World Puzzle Championship and puzzle enthusiast communities online. It has been included in multiple WPC events and is well-regarded among competitive puzzlers for its clean logic and visual elegance. The puzzle also appeared in Simon Tatham's Portable Puzzle Collection, an influential open-source puzzle suite, which introduced it to a wide audience of casual players.

The puzzle has a natural connection to algorithmic problems in computer science — partitioning a grid into rectangles is related to rectangle packing and covering problems, which have applications in VLSI chip design, image segmentation, and computational geometry. This mathematical underpinning gives Shikaku a depth that appeals to both puzzle solvers and mathematicians.