Magic constant
Encyclopedia
The magic constant or magic sum of a magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

 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 cube
Magic cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...

s.

Normal magic squares

If a magic square of order n is normal
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

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

This formula is a consequence of the formula for the sum of the first n integers
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
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...

):
15
15 (number)
15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name....

, 34
34 (number)
34 is the natural number following 33 and preceding 35.-In mathematics:34 is the ninth distinct semiprime and has four divisors including unity and itself. Its neighbors, 33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers...

, 65
65 (number)
65 is the natural number following 64 and preceding 66.-In mathematics:Sixty-five is the 23rd semiprime and the 3rd of the form it is an octagonal number. It is also a Cullen number...

, 111
111 (number)
111 is the natural number following 110 and preceding 112. It is the lowest positive integer requiring six syllables to name in American English, or seven syllables in Canadian and British English...

, 175
175 (number)
175 is the natural number following 174 and preceding 176.-In mathematics:* 175 is an odd number* 175 is a composite number* 175 is a decagonal number, a figurate number that represents a decagon* 175 is a deficient number, as 73 is less than 175...

, 260
260 (number)
260 is the magic constant of the n×n normal magic square and n-queens problem for n = 8, the size of an actual chess board....

, 369
369 (number)
Three hundred sixty-nine is the natural number following three hundred sixty-eight and preceding three hundred seventy.- In mathematics :369 is the magic constant of the 9 × 9 magic square and the n-Queens Problem for n = 9.There are 369 free octominoes .-In other...

, 505, 671, 870, ….


The numbers in any row, column, or diagonal of a normal magic square form a magic series
Magic series
A magic series is a set of distinct positive numbers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts....

.

Magic cubes

Similarly, if a magic cube consists of the numbers 1, 2, ..., n³, then it has magic constant

Magic tesseracts

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

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

Magic stars

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

External links

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