Divide by Squares

"Divide by Squares" is a popular logic puzzle, also commonly known in English as "Shikaku" or "Divide into Cells".

These names accurately capture the essence of the game: players are required to divide the game grid into several rectangular (or square) regions. Each region must contain a single number, and the value of that number must equal the number of cells within that region. While the rules of "Divide by Squares" are simple to understand, mastering the game and solving puzzles quickly can be quite challenging, which is a characteristic of many Japanese puzzles.

Game History

"Divide by Squares" originated in Japan, where it is called "四角に切れ (Shikaku ni kire)". The game was first published by the Japanese logic puzzle magazine "Nikoli," which began publishing columns dedicated to logic puzzles in the late 1980s.

From 1989 to 1999, Nikoli magazine published hundreds of unique logic games that were subsequently modified, improved, and renamed. The designers of these games included Nikoli staff members and numerous contributing readers.

Kaji Maki, the founder of Nikoli magazine, pointed out that the published puzzles shared a common characteristic: a clear grading of difficulty, from the simplest beginner levels to the most complex expert levels. This type of difficulty scaling is common in Japan, where it's believed that everyone should start at the bottom and gradually advance their skills. Therefore, increasing the difficulty of puzzles (including logic and computer-based ones) is also a significant Japanese invention.

Notably, many famous games published in Nikoli and popular worldwide are not credited to specific authors. Currently, only the pen names and genders of the contributing readers are known. It's estimated that approximately 80% of the published game creators are male.

To learn more about these puzzles, one way is to attend the annual "Nikoli Party" held in Tokyo. This event brings together the magazine's staff and invited guests, including some of the logic puzzle authors.

The "Divide by Squares" game is a rare case where we know not only the author's pen name but also his real name. The author is Yoshinao Yasufuku, a reader of Nikoli magazine, whose name has become a part of puzzle history. Although "Divide by Squares" was initially designed purely for entertainment, it is now often used as an educational tool in mathematics. In many schools, it is used to demonstrate the following mathematical concepts:

Prime numbers;
Factors (or divisors);
Square roots;
Perfect squares;
Area of a rectangle;
Area of a square.

Despite its simple rules, "Divide by Squares" effectively demonstrates at least 6 important mathematical concepts, such as: inclusion, disjunction, union, intersection, bijection, and section. From a mathematical perspective, "Divide by Squares" does hold a certain value, though this might only be meaningful to those passionate about mathematics. For others, "Divide by Squares" remains an excellent way to pass the time and exercise logical thinking skills.

Try "Divide by Squares" now (free and without registration) – you’ll surely fall in love with this game!

"Divide by Squares" is a logic puzzle played on a rectangular grid, often a square grid. The larger the grid, the more difficult it is for players to find the correct solution.

Like many other Nikoli puzzles, "Divide by Squares" is categorized by difficulty. Even children can solve simple puzzles, but even for adults, solving complex ones can take a significant amount of time. However, if you understand the rules and learn some solving strategies, the process becomes much easier.

Basic Rules

The rules of "Divide by Squares" puzzles are simple and can be understood in minutes. On a rectangular grid divided into square cells, there are numbers whose values are always less than or equal to the total number of cells in the grid. The player's goal is to enclose each number with a rectangle, so that the number of cells within each rectangle is equal to the number inside that rectangle. For example, the number 4 should be enclosed in a rectangle with 4 cells, the number 7 should be enclosed in a rectangle with 7 cells, and so on.

The basic rules players need to remember for "Divide by Squares" include:

Each rectangle can contain only one number.
The number of cells within a rectangle must exactly match the number inside that rectangle.
Rectangles cannot overlap.
At the end of the game, all cells on the grid must be allocated to a rectangle; no cell should remain uncovered.

In "Divide by Squares", only simple rectangular shapes are allowed; complex shapes with protrusions are not permitted. For example, 1x2, 2x2, 1x7, and 2x5 are valid rectangles. The value of each rectangle (i.e., the number of cells it contains) must strictly equal the product of its length and width, which is the area of the rectangle. For example, a 1x2 rectangle has an area of 2, a 2x2 rectangle has an area of 4, and a 1x7 rectangle has an area of 7.

How to Solve Puzzles

After mastering the basic rules of "Divide by Squares", you can start solving puzzles. You should begin with smaller grids and then progress to larger ones (such as 10x10 or larger). Regardless, the solving strategy comes down to finding and identifying the correct rectangles to enclose each number on the grid.

The simplest and most straightforward case is the number "1." You can immediately enclose these cells with rectangles. However, for other numbers, you'll need to think carefully.

To solve "Divide by Squares" quickly, follow these tips:

Rectangles containing prime numbers always have a width of 1. Prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19, etc. That is, numbers that are only divisible by 1 and themselves. It's impossible to construct a rectangle with a width greater than 1 using a prime number!
If a number is a perfect square (e.g., 4, 9, 16, 25, 36), you can form a square around it with a side length equal to the square root of that number. For example, the number 4 can form a 2x2 square, and the number 25 can form a 5x5 square.
If a number has multiple factors (divisors), the size of the rectangle containing that number may have several options (its width might not be 1).

In small grids, most rectangles are usually elongated and only one or two cells wide. Squares are less common, and rectangles with a width greater than two cells are even rarer. The player's task is to find clear partition solutions and draw the borders around them. This requires logical thinking and the ability to use mathematical reasoning.

After playing a few games of "Divide by Squares", you'll find that this simple game is so engaging that you'll want to play it again and again.

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Divide by Squares