M

A magic square of order n is an n x n array containing the integers 1, 2, 3 ... n². 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+...+n² = n²(n²+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(n²+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	11

Let E = {e₁, e₂, ...e_m} be a finite set, and let F be a family of subsets of E: then F is a matroid if it satisfies

• e_i in F for each i,
• if G is in F, and if H is a non-empty subset of G, then H is in F.
• for each S that is a subset of E, if G and H are two members of F contained in S and maximal with this property, then |G|=|H|.

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 }.

A Monoid is almost a group , it only fails to be a group because every element in a monoid need not have an inverse. A monoid does have an identity element (contrast this with a semigroup ).