A heterosquare of order n is an arrangement of the integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s 1 to n² in a square, such that the rows, columns, and diagonals all sum to different values. There are no heterosquares of order 2, but heterosquares exist for any order n ≥ 3.

Order 3	Order 4	Order 5

Heterosquares are easily constructed, as shown in the above examples. If n is odd, filling the square in a spiral

Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

 pattern will produce a heterosquare. And if n is even, a heterosquare results from writing the numbers 1 to n² in order, then exchanging 1 and 2.

It is strongly suspected that there are exactly 3120 essentially different

Frénicle standard form

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:# the element at position [1,1] is the smallest of the four corner elements; and...

 heterosquares of order 3.

An antimagic square

Antimagic square

An antimagic square of order n is an arrangement of the numbers 1 to n2 in a square, such that the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers...

is a special kind of heterosquare where the 2n + 2 row, column and diagonal sums are consecutive integers.