If the whole numbers used are consecutive from 1 to *n*², 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.

### Property

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: \(S\space = \space \dfrac{n(n^2+1)}{2}\).

### Examples

- 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.