### 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', Magic birthday square 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.

[panmagic]  A magic square is panmagic if the addition of the numbers 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.

[inverse (or complementary)]  If you replace the highest number by the lowest number, the second highest number by the second lowest number and so on, than you get the inverse (= complementary) magic square. The inverse magic square has the same magic features as the original magic square. A special kind of inverse (complementary) is symmetric (= self complementary).

[symmetric (or self complementary)]  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)').

N.B.: If you replace the numbers in a symmetric (= self complementary) magic square by the inverse (= highest in stead of lowest, second highest in stead of second lowest, ...) numbers, than you get the same magic square, which is rotated by 180 degrees (= up-site down).

[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.

N.B.1: Compact can also mean that only the numbers in the four corners of 3x3, 4x4, 5x5, ... give the same (part of the magic) sum. I don't use this defenition on my website.

N.B.2: On this website you find also magic squares which are AxB compact. See for example the 12x12 composite magic square, which is 3x4 (and 4x3) compact.

[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) also 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.

N.B.: Also for order is multiple of 4 from 8x8 and up, you can get a ultramagic solution, so the (8x8, 12x12, 16x16, ...) magic square is panmagic, symmetric and 2x2 compact. See basic key method ultra magic.

[Franklin magic]  Benjamin Franklin constructed 8x8 and 16x16 (semi) magic squares, which are 2x2 compact and half rows and haf columns give half of the magic sum. The diagonals of a Franklin magic square don't give the magic sum, but the bent [parallel] [mirrored] diagonals give the magic sum. See famous magic squares.

[Franklin panmagic]  A Franklin panmagic square is 2x2 compact, panmagic and half rows/columns/diagonals give half of the magic sum.

N.B.: A Franklin panmagic square is possible for each order which is a multiple of 4 from 8x8, with exception of the 12x12 magic square. If the order is a multiple of 8 it is easy to get a Franklin panmagic square. If the order is an odd multiple of 4 from 20x20 it is not easy to get a Franklin panmagic square (you need computer programming).

[Complete]  A magic square (order is multiple of 4) is complete under the following conditions. Firstly the magic square must be 2x2 compact and panmagic. Secondly - if you split the diagonals in two parts - the first number in the second half must be the inverse of the first number in the first half. The same should be true for the second, third, fourth, ... number (see explanation most perfect).

N.B.: According to Kathleen Ollernshaw (the famous mathematician) the complete magic square is most perfect. Willem Barink and I disagree and we think most perfect magic squares have a different, more tight structure; see below.

[most perfect]  A most perfect magic square on this website consists of (1x1,) 2x2, 3x3, 4x4, ... proportional panmagic 4x4 (sub)squares, which must be connected adequately, so the magic square is fully 2x2 compact (see that sometimes corrections are necessary to get adequate connections; see for example the Khajuraho method).  If a magic square has the above mentioned (most perfect) structure each 1/2, 1/3, 1/4, 1/5, or ... rows/columns/diagonals give 1/2, 1/3, 1/4, 1/5, ... of the magic sum. See also explanation most perfect.

N.B.1: According to above mentioned definition the 8x8 Franklin panmagic square is most perfect. For order is multiple of 8 from 16x16, most perfect is more than Franklin panmagic (because 1/4 in stead of 1/2 rows/columns/diagonals give 1/4 of the magic sum).

N.B.2: The complete magic square is a transformation of the above mentioned most perfect magic square. On this website you can find in each download of a most perfect magic square the transformation into a complete magic square (by swapping rows and columns).

[Magic feature X]  We have learned from Willem Barink that a part of the most perfect magic squares has the extra magic feature X, meaning that in each row and column number 1+2 = 5+6 = 9+10 = 13+14, ... & number 3+4 = 7+8 = 11+12 = 15+16, ...; see also  explanation most perfect.

[The perfect magic square]  A perfect magic square is possible for order is multiple of 4 from 8x8. The perfect magic square is not only most perfect with the extra magic feature X. But watching from the 4x4 subsquares (starting in one of the four corners), you can find all numbers (from 1 up to nxn) in sequence. See i.e. the perfect (16x16) magic square.

[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 2x2 inverse numbers, but not the middle numbers), and you can put borders around it again and again. For example a concentric 14x14 magic square consists of a (each time proportional) 4x4 in 6x6 in 8x8 in 10x10 in 12x12 in 14x14 magic square.

[Inlaid magic square]  An inlaid magic square is a magic square wherein one or more magic subsquares are found. See for example the 22x22 inlaid magic square with 4x4 and 7x7 subsquares within the 22x22 magic square.

[p-multimagic square]  There are 2-mulimagic (= bimagic), 3-multimagic (= trimagic), 4-multimagic (= tetramagic), 5-multimagic (= pentamagic), ... squares. A bimagic square remains magic even if all its numbers are replaced by their squares. So trimagic means that you can replace the numbers by their squares and third powers; tetramagic means that you can replace the numbers by their squares, third powers and fourth powers; pentamagic means that you can replace the numbers by their squares, third powers, fourth powers and fifth powers; ... On this website you find a 12x12 trimagic square and bimagic squares for order 8, 9, 16, 25 and 32. It is difficult to analyze a n-multimagic square. I have succeeded in analyzing the 25x25 bimagic square.

[centre]  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.

[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 than 1 that can only be divided by itself and 1.