Magic Square

Magic Square

Table of whole numbers arranged in a square (3 × 3, 4 × 4, etc.) so that the sum of the numbers vertically, horizontally, and diagonally is always the same.

If the whole numbers used are consecutive from 1 to n², we call this a normal magic square of n by n.

The sum of the number on the same line or in the same column or on a diagonal is called the magic sum or the density of the magic square.

We sometimes use the term heterogeneous square or heterosquare for a square grid of numbers where the sum of the number situated vertically, horizontally, or diagonally, is never the same.

If a heterosquare is formed by a sequence of consecutive whole numbers from 1 to \(n^2\), we call it a normal heterosquare.

We also sometimes use the term antimagic square for a heterosquare where the sums of the numbers found on the lines, diagonals, and columns form a sequence of whole numbers.


To calculate the magic sum S of a magic square formed by numbers from 1 to \(n\) including \(n^2\) cases, we can use the formula: \(S\space = \space \dfrac{n(n^2+1)}{2}\).


  • A normal 3 × 3 magic square with a magic sum equal to 15:
  • A normal heterosquare:
  • An antimagic square: Here, the sums of the lines, columns, and diagonals form a sequence of whole numbers from 29 to 38.

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