Editing Magic square (section)

==Properties of magic squares==
===Magic constant===
{{main|Magic constant}}
The constant that is the sum of any row, or column, or diagonal is called the [[magic constant]] or magic sum, ''M.'' Every normal magic square has a constant dependent on the order {{mvar|n}}, calculated by the formula <math>M = n(n^2 + 1)/2</math>. This can be demonstrated by noting that the sum of <math>1,2,...,n^2</math> is <math>n^2(n^2 + 1)/2</math>. Since the sum of each row is <math>M</math>, the sum of <math>n</math> rows is <math>n M = n^2(n^2 + 1)/2</math>, which when divided by the order {{mvar|n}} yields the magic constant as <math>M = n(n^2 + 1)/2</math>. 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]]).

===Magic square of order 1 is trivial===
The 1×1 magic square, with only one cell containing the number 1, is called ''[[Triviality (mathematics)|trivial]]'', because it is typically not under consideration when discussing magic squares; but it is indeed a magic square by definition, if a single cell is regarded as a square of order one.

===Magic square of order 2 cannot be constructed===
Normal magic squares of all sizes can be constructed except 2×2 (that is, where order ''n'' = 2).<ref>{{cite web | url = http://mathforum.org/alejandre/magic.square/adler/adler5.html | archive-url = https://web.archive.org/web/20180302092216/http://mathforum.org/alejandre/magic.square/adler/adler5.html | url-status = dead | archive-date = 2018-03-02 | title = Why there are no 2x2 magic squares | website = mathforum.org | first1 = Allan | last1 = Adler | first2 = Suzanne | last2 = Alejandre }}</ref>

===Center of mass===
If the numbers in the magic square are seen as masses located in various cells, then the [[center of mass]] of a magic square coincides with its geometric center.

===Moment of inertia===
The ''moment of inertia'' of a magic square has been defined as the sum over all cells of the number in the cell times the squared distance from the center of the cell to the center of the square; here the unit of measurement is the width of one cell.<ref name=Loly/> (Thus for example a corner cell of a 3×3 square has a distance of <math>\sqrt{2},</math> a non-corner edge cell has a distance of 1, and the center cell has a distance of 0.) Then all magic squares of a given order have the same moment of inertia as each other. For the order-3 case the moment of inertia is always 60, while for the order-4 case the moment of inertia is always 340. In general, for the ''n''×''n'' case the moment of inertia is <math>n^2(n^4-1)/12.</math><ref name=Loly/>

===Birkhoff–von Neumann decomposition===
Dividing each number of the magic square by the magic constant will yield a [[doubly stochastic matrix]], whose row sums and column sums equal to unity. However, unlike the doubly stochastic matrix, the diagonal sums of such matrices will also equal to unity. Thus, such matrices constitute a subset of doubly stochastic matrix. The Birkhoff–von Neumann theorem states that for any doubly stochastic matrix <math>A</math>, there exists real numbers <math>\theta_1,\ldots,\theta_k \ge 0</math>, where <math>\sum_{i=1}^k \theta_i = 1</math> and [[permutation matrices]] <math>P_1,\ldots,P_k</math> such that

:<math>A = \theta_1 P_1 + \cdots + \theta_k P_k. </math>

This representation may not be unique in general. By Marcus-Ree theorem, however, there need not be more than <math> k \le n^2 - 2n + 2</math> terms in any decomposition.<ref>{{cite journal|last1=Marcus|first1=M.|last2=Ree|first2=R.|title=Diagonals of doubly stochastic matrices|journal=The Quarterly Journal of Mathematics|date=1959|volume=10|issue=1|pages=296–302|doi=10.1093/qmath/10.1.296}}</ref> Clearly, this decomposition carries over to magic squares as well, since a magic square can be recovered from a doubly stochastic matrix by multiplying it by the magic constant.