Magic constant

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15.

The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes.

Normal magic squares

If a magic square of order n is normal (i.e., it contains the numbers 1 to n²), then the magic constant depends only on n; its value is

$M_2(n) = \frac{n(n^2+1)}{2}.$

This formula is a consequence of the formula for the sum of the first n integers

$1 + 2 + ... + k = \frac{k(k+1)}{2}$

applied to the case k = n², yielding n²(n²+1)/2, which is then divided by n because there are n rows, all of which sum to the same value.

The magic constants of normal magic squares of order n = 3, 4, 5, … are (sequence A006003 in OEIS):

15, 34, 65, 111, 175, 260, 369, 505, 671, 870, ….

The numbers in any row, column, or diagonal of a normal magic square form a magic series.

Magic cubes

Similarly, if a magic cube consists of the numbers 1, 2, ..., n³, then it has magic constant (sequence A027441 in OEIS)

$M_3(n) = \frac{n(n^3+1)}{2}.$

Magic tesseracts

In 4 dimensions, a magic tesseract consists of the numbers 1, 2, ..., n⁴, and has magic constant

$M_4(n) = \frac{n(n^4+1)}{2}.$

In general, for dimension d and order n, a magic hypercube consists of the numbers 1, 2, ..., n^d, and has the magic constant

$M_d(n) = \frac{n(n^d+1)}{2}.$

Magic stars

The magic constant of an n-pointed normal magic star is M = 4n + 2.

See also

• Magic number (physics)

External links

• 260 as magic constant for 8-queens problem and 8x8 magic square
• Hypercube Math formulae