Editing Magic constant (section)

{{For|unnamed numerical constants|Magic number (programming)#Unnamed numerical constants}}

The '''magic constant''' or '''magic sum''' of a [[magic square]] is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is, a magic square which contains the numbers 1, 2, ..., ''n''<sup>2</sup> – the magic constant is <math>M = n \cdot \frac{n^2 + 1}{2}</math>.
[[File:Magicsquareexample.svg|center]]

For normal magic squares of orders ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence [[OEIS:A006003|A006003]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).
For example, a normal 8&thinsp;×&thinsp;8 square will always equate to 260 for each row, column, or diagonal.
The normal magic constant of order ''n'' is {{sfrac|''n''<sup>3</sup> + ''n''|2}}.
The largest magic constant of normal magic square which is also a:
*[[triangular number]] is [[15 (number)|15]] (solve the Diophantine equation {{nowrap|1=''x''<sup>2</sup> = ''y''<sup>3</sup> + 16''y'' + 16,}} where ''y'' is divisible by 4);
*[[square number]] is [[1 (number)|1]] (solve the Diophantine equation {{nowrap|1=''x''<sup>2</sup> = ''y''<sup>3</sup> + 4''y'',}} where ''y'' is even);
*[[generalized pentagonal number]] is 171535 (solve the Diophantine equation {{nowrap|1=''x''<sup>2</sup> = ''y''<sup>3</sup> + 144''y'' + 144,}} where ''y'' is divisible by 12);<!-- This can be considered Original Research [[WP:OR]] -->
*[[tetrahedral number]] is 2925.

Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares.

However, there are infinitely many rational triangular numbers, rational generalized pentagonal numbers and rational tetrahedral numbers which are also magic constants of rational order.

The term '''magic constant''' or '''magic sum''' is similarly applied to other "magic" figures such as [[magic star]]s and [[magic cube]]s. Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant.<ref name="Polyiamond Magic Constant">{{Cite web|url=http://oeis.org/A303295/|title=A303295 - Oeis}}</ref>