A magic square of order n is an n x n array containing the integers 1, 2, 3 ... n^{2}. The sum of every row, column, and the two principal diagonals is the same number s, called the magic sum. The magic sum can be deduced; the sum of all entries in the magic square is counted in two ways, first straightforwardly -- 1+2+3+...+n^{2} = n^{2}(n^{2}+1)/2 then by adding up the rows (or columns) there are n rows (or columns) each of which sums to s, thus the sum of all entries in the magic square is also ns. Equating these quantities, and solving for s yields:
s = n(n^{2}+1)/2
Here is a magic square of order 5:
15 8 1 24 17 16 14 7 5 23 22 20 13 6 4 3 21 19 12 10 9 2 25 18 11The magic sum is 65.
Let E = {e_{1}, e_{2}, ...e_{m}} be a finite set, and let F be a family of subsets of E: then F is a matroid if it satisfies
The boundaries of the cells of the Voronoi diagram of the edges of a simple polygon.
In a vector space, the Minkowski sum of sets A and B is the set { a + b | a in A, b in B }.
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