Research Activities > Research Introduction > What is a Magic Square?

What is a Magic Square?

What is a Magic Square?

A magic square is an arrangement of numbers in a square grid such that every possible way you sum the numbers in a row, column, or along a diagonal, the sum will come out to be the same. In particular, it refers to grids that have numbers filling the grids in columns n, and row n (n x n) with numbers from 1 to n2.


3×3 magic square

The most famous magic square is the grid displayed on the right with rows and columns with 3 squares each (3×3). In the case of the magic square on the right, the sums of the rows, columns, or diagonals each come out to 15. In a 3×3 magic square, excluding reflections or rotations, this is the only possible formation.
The history of magic squares is long, and there are references in the “scroll of the river Lo” (or Lo Shu) in Chinese astrology and in Western numerology as the Magic Square of Saturn.

To this day, it is famous as a mathematical puzzle, and this ancient 3×3 puzzle is still popular and takes only simple mathematics to solve.


The various types of magic squares


Though we use the term Magic Square, there are actually many kinds. Moments ago it was mentioned that numbers from 1 to n2 are used, but there are magic squares using random numbers or even letters. Furthermore, there are also other types which, instead of seeking a sum, may use multiplications or even powers. There are also other shapes besides squares, including circles, or even star-shaped versions, and some that do not use numbers at all.

Even the popular puzzle game called Sudoku is a type of magic square known as the Latin Square.The image on the right is a magic square of a four-character phrase. If you are able to understand the phrase see if you can figure out what are the rules for it.

In our lab, we treat “any element within a grid which follows a law” as “general Magic Squares.”

What we research using magic squares in our lab

At Ishida lab, we create and analyze new types of Magic Squares using new laws.

  • Research on Dynamic Magic Squares

We research magic squares with new rules obtained as it follows “rules which change one magic square to ‘another magic square.’” We think of a variety of dynamic structures based on the nature of the magic squares and the defined rules for change.

  • Research on the SMP (Stable Marriage Problem) Magic Squares

This is a magic square which has taken in the selection order from an SMP as its rules. Magic squares with different qualities appear based on the selection order which it follows as its rule.

  • Research on Sudoku (Number-Placing)

Sudoku is a 9×9 semi-filled pencil puzzle in which you fill the blank spaces in the rows, columns and blocks without duplication of each numbers. Sudoku is generally considered a type of magic square. The level of difficulty changes depending on how the original numbers are placed, and we research these various levels of difficulty.

Research Activities > Research Introduction > What is a Magic Square?

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