Editing Magic square (section)

==Classification of magic squares==
[[File:4x4_magic_square_hierarchy.svg|thumb|upright|[[Euler diagram]] of the properties of some types of 4×4 magic squares. Cells of the same colour sum to the magic constant.<br /><nowiki>*</nowiki> In 4×4 most-perfect magic squares, any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant.]]

While the classification of magic squares can be done in many ways, some useful categories are given below. An ''n''×''n'' square array of integers 1, 2, ..., ''n''<sup>2</sup> is called:
* ''Semi-magic square'' when its rows and columns sum to give the magic constant.
* ''Simple magic square'' when its rows, columns, and two diagonals sum to give magic constant and no more. They are also known as ''ordinary magic squares'' or ''normal magic squares''.
* ''Self-complementary magic square'' when it is a magic square which when complemented (i.e. each number subtracted from ''n''<sup>2</sup> + 1) will give a rotated or reflected version of the original magic square.
* ''[[Associative magic square]]'' when it is a magic square with a further property that every number added to the number equidistant, in a straight line, from the center gives ''n''<sup>2</sup> + 1. They are also called ''symmetric magic squares''. Associative magic squares do not exist for squares of singly even order. All associative magic square are self-complementary magic squares as well.
* ''[[Pandiagonal magic square]]'' when it is a magic square with a further property that the broken diagonals sum to the magic constant. They are also called ''panmagic squares'', ''perfect squares'', ''diabolic squares'', ''Jain squares'', or ''Nasik squares''. Panmagic squares do not exist for singly even orders. However, singly even non-normal squares can be panmagic. 
* ''Ultra magic square'' when it is both associative and pandiagonal magic square. Ultra magic square exist only for orders ''n'' ≥ 5.
* ''Bordered magic square'' when it is a magic square and it remains magic when the rows and columns on the outer edge are removed. They are also called ''concentric bordered magic squares'' if removing a border of a square successively gives another smaller bordered magic square. Bordered magic square do not exist for order 4.
* ''Composite magic square'' when it is a magic square that is created by "multiplying" (in some sense) smaller magic squares, such that the order of the composite magic square is a multiple of the order of the smaller squares. Such squares can usually be partitioned into smaller non-overlapping magic sub-squares. 
* ''Inlaid magic square'' when it is a magic square inside which a magic sub-square is embedded, regardless of construction technique. The embedded magic sub-squares are themselves referred to as ''inlays''.
* ''[[Most-perfect magic square]]'' when it is a pandiagonal magic square with two further properties (i) each 2×2 subsquare add to 1/''k'' of the magic constant where ''n'' = 4''k'', and (ii) all pairs of integers distant ''n''/2 along any diagonal (major or broken) are complementary (i.e. they sum to ''n''<sup>2</sup> + 1). The first property is referred to as ''compactness'', while the second property is referred to as ''completeness''. Most-perfect magic squares exist only for squares of doubly even order. All the pandiagonal squares of order 4 are also most perfect.
* ''Franklin magic square'' when it is a doubly even magic square with three further properties (i) every bent diagonal adds to the magic constant, (ii) every half row and half column starting at an outside edge adds to half the magic constant, and (iii) the square is ''compact''.
* ''[[Multimagic square]]'' when it is a magic square that remains magic even if all its numbers are replaced by their ''k''-th power for 1 ≤ ''k'' ≤ ''P''. They are also known as ''P-multimagic square'' or ''satanic squares''. They are also referred to as ''bimagic squares'', ''trimagic squares'', ''tetramagic squares'', and ''pentamagic squares'' when the value of ''P'' is 2, 3, 4, and 5 respectively.