Features of magic squares


[pure]   A magic square is pure if it consists of the numbers 1 up to n x n. A pure magic 3x3 square consists of the numbers 1 up to (3 x 3 =) 9. On this website you find only pure magic squares with the exception of 'Each magic sum 4x4' or 'Each magic sum 5x5'

[minimal magic features] 
 The addition of the numbers of each row/column/ diagonal must give the same magic sum.

[magic sum] 
For each pure magic square you can calculate the magic sum. The magic sum is [(1 + n x n) / 2] x n. For example the sum of the 3x3 magic square is: [(1 + 3 x 3) / 2) x 3 = 15.


[concentric]  An odd concentric magic square consists of a centre of one cell (with the middle number in it) and an even concentric magic square consists of a centre of 2x2 cells (with the middle 4 numbers in it), and you can put borders around it again and again. For example a concentric magic 14x14 square consists of a (each time proportional) 4x4 in 6x6 in 8x8 in 10x10 in 12x12 in 14x14 magic square. 


[panmagic]  A magic square is panmagic if the addition of the digits of each pandiagonal gives the magic sum. A pandiagonal is a broken diagonal, which consists of two parts. The first part is a line, which starts from the outside row or outside column (but not from a corner) of the magic square. The second part is a line or a dot (and the dot ends in one of the corners of the magic squares). See for example the pandiagonals of the panmagic 4x4 square.


[symmetric]  In a symmetric magic square addition of two numbers, which can be connected with a straight line through the centre of the magic square and which are at the same distance from the centre, gives the same sum. The sum is 1 + n x n (for example the sum in a symmetric 5x5 magic square is: 1 + 5 x 5 = 26). It is also possible that the magic square is not symmetric as a whole, but the magic square is symmetric in each sub-square (for example 'basic key method (ultra magic)'). 

The centre of an odd magic square is the middle cell (n.b.: in the middle cell of an odd symmetric magic square you always find the middle number; for example in a symmetric 5x5 magic square you find the number 13 in the middle cell). The centre of an even magic square is the crosspoint of the middle 2x2 cells.


[compact]  If a magic square is a multiple of 2, 3, 5, 7, … than compact means, that each randomly chosen 2x2, 3x3, 5x5, 7x7, … sub-square gives the same (proportional part of the) magic sum. A magic square can be double compact. For example the ultra magic 15x15 square on this website gives a proportional part of the magic sum for each 3x3 sub- square and for each 5x5 sub-square. 


[ultramagic]  For an odd order (with exception of the 3x3 magic square) is ultra magic the most magic square. An odd ultra magic square is always panmagic and symmetric and (if the order of the square is not a prime number) compact. If possible in the ultra magic square also a part of each row, column and/or diagonal gives a proportional part of the magic sum. For example in the ultra panmagic 27x27 square on this website gives each 1/9 row, 1/9 column and 1/3 diagonal a proportional part of the magic sum.


[most perfect]  For orders which are a multiple of four the most perfect (Franklin pan)magic square is the most magic square. Willem Barink taught us, that a little part of the most perfect magic squares has an extra tight structure.

[Scope]  Scope means, the (absolute or relative) number of different magic squares you can produce by using a method of construction. A scope of 100% means, that you can produce all possible magic squares by using a method of construction. For example with the shift method you can produce all odd panmagic squares which are not a multiple of 3.

[prime number 
A prime number is a whole number greater thawn 1 that can only be divided by itself and 1.