Order5 Pandiagonal Magic Squares

There are 36 ‘essentially different’ order5 pandiagonal magic squares that can each be transformed into 3 other magic squares. The resulting 144 pandiagonal magic squares can each in turn be transformed cyclically to 24 other magic squares by successively moving a row or column from 1 side of the square to the other side. Completing these transformations on all 36 essentially different magic squares will produce the complete set of 3600 pandiagonal magic squares of order5. Each of the 36 essentially different magic squares is transformed to 3 others as follows.
The total number of order5 pandiagonal basic magic squares is 36 times 4 times 25 equals 3600.
Because using a standard notation is so important when comparing magic square lists, I include here the definitions for two terms I will be using on this page: Normalizing.............Rotating and /or reflecting a magic square or magic star to achieve the standard position so the figure may be assigned an index number. Standard position....Any magic square may be disguised to make 7 other (apparently) different magic squares by means of rotations and reflections. These variations are NOT considered as new magic squares for purposes of enumeration. For the purpose of listing and indexing magic squares, a standard position must be defined. The magic square is then rotated and/or reflected until it is in this position. This position was defined by Frénicle in 1693 and consists of only two requirements.
An essentially different order5 magic square is defined as follows ( Benson & Jacoby).
All references to the number of magic squares refers to the basic squares which is normally understood in magic square discussions. Alan Grogono, in contrast, uses the mathematical approach and considers the total number of squares (which includes rotations and reflections).
When making the above transformations, the resulting magic square will not necessarily be normalized. That is, the second cell in the top row may not be smaller then the first cell in the second row. If required, this may simply be done by reflecting the square around the leading diagonal (exchanging rows and columns). No rotation will be required because the top left hand cell will always contain the number 1. It seems that if the original square is basic, about 4 of the 24 squares resulting from the cyclical transformation will be basic and the other 20 will require normalizing. If the original square is not basic, none of the other 24 will be either. A comparison of the following derivatives from square # 31 with those above
from square # 1 will show that different squares require normalization to
produce a basic magic square.
After the cyclical permutations are performed, again many squares will require normalization. Because different lists of the 144 fundamental magic squares may be compiled differently, lists by different persons will very likely not match. Regardless of which list of the 144 fundamental squares is used, after the cyclical permutations are performed, and all resulting squares are normalized (where required), the resulting list of sorted 3600 pandiagonal magic squares will be identical. This suggests that an intermediate normalization step may be dispensed with, and only the final normalization be performed, where required, on the finished list. Alan Grogono refers to these 144 squares as ‘unique’. (He uses the number series from 0 to 24 instead of 1 to 25 to simplify mathematical computing.) http://www.grogono.com/magic/text5x5pan144.shtml#Subtop Mutsumi Suzuki refers to them as "fundamental’. His pages are now found in the MathForum The following 36 essentially different magic squares were compiled from
tables in List of 36 Essentially different The left hand number above each square is the Benson & Jacoby sequence
number. 1 2 35 2 4 31 3 1 24 4 3 20 5 6 33 6 8 27 1 7 13 19 25 1 7 13 20 24 1 7 13 24 20 1 7 13 25 19 1 7 14 18 25 1 7 14 20 23 14 20 21 2 8 15 19 21 2 8 14 25 16 2 8 15 24 16 2 8 13 20 21 2 9 15 18 21 2 9 22 3 9 15 16 22 3 10 14 16 17 3 9 15 21 17 3 10 14 21 22 4 8 15 16 22 4 10 13 16 10 11 17 23 4 9 11 17 23 5 10 11 22 18 4 9 11 22 18 5 10 11 17 24 3 8 11 17 24 5 18 24 5 6 12 18 25 4 6 12 23 19 5 6 12 23 20 4 6 12 19 23 5 6 12 19 25 3 6 12 7 5 22 8 7 16 9 10 29 10 12 25 11 9 18 12 11 14 1 7 14 23 20 1 7 14 25 18 1 7 15 18 24 1 7 15 19 23 1 7 15 23 19 1 7 15 24 18 13 25 16 2 9 15 23 16 2 9 13 19 21 2 10 14 18 21 2 10 13 24 16 2 10 14 23 16 2 10 17 4 8 15 21 17 4 10 13 21 22 5 8 14 16 22 5 9 13 16 17 5 8 14 21 17 5 9 13 21 10 11 22 19 3 8 11 22 19 5 9 11 17 25 3 8 11 17 25 4 9 11 22 20 3 8 11 22 20 4 24 18 5 6 12 24 20 3 6 12 20 23 4 6 12 20 24 3 6 12 25 18 4 6 12 25 19 3 6 12 13 14 34 14 13 30 15 15 12 16 16 8 17 18 36 18 17 26 1 7 18 14 25 1 7 18 15 24 1 7 18 24 15 1 7 18 25 14 1 7 19 13 25 1 7 19 15 23 13 24 5 6 17 13 25 4 6 17 19 25 11 2 8 20 24 11 2 8 14 23 5 6 17 14 25 3 6 17 10 16 12 23 4 9 16 12 23 5 12 3 9 20 21 12 3 10 19 21 10 16 12 24 3 8 16 12 24 5 22 3 9 20 11 22 3 10 19 11 10 16 22 13 4 9 16 22 13 5 22 4 8 20 11 22 4 10 18 11 19 15 21 2 8 20 14 21 2 8 23 14 5 6 17 23 15 4 6 17 18 15 21 2 9 20 13 21 2 9 19 19 10 20 20 4 21 22 32 22 21 28 23 23 6 24 24 2 1 7 19 23 15 1 7 19 25 13 1 7 20 13 24 1 7 20 14 23 1 7 20 23 14 1 7 20 24 13 18 25 11 2 9 20 23 11 2 9 15 23 4 6 17 15 24 3 6 17 18 24 11 2 10 19 23 11 2 10 12 4 8 20 21 12 4 10 18 21 9 16 12 25 3 8 16 12 25 4 12 5 8 19 21 12 5 9 18 21 10 16 22 14 3 8 16 22 14 5 22 5 8 19 11 22 5 9 18 11 9 16 22 15 3 8 16 22 15 4 24 13 5 6 17 24 15 3 6 17 18 14 21 2 10 19 13 21 2 10 25 13 4 6 17 25 14 3 6 17 25 26 21 26 25 17 27 28 9 28 27 5 29 30 23 30 29 13 1 7 23 14 20 1 7 23 15 19 1 7 23 19 15 1 7 23 20 14 1 7 24 13 20 1 7 24 15 18 13 19 5 6 22 13 20 4 6 22 18 14 5 6 22 18 15 4 6 22 14 18 5 6 22 14 20 3 6 22 10 21 12 18 4 9 21 12 18 5 10 21 17 13 4 9 21 17 13 5 10 21 12 19 3 8 21 12 19 5 17 3 9 25 11 17 3 10 24 11 12 3 9 25 16 12 3 10 24 16 17 4 8 25 11 17 4 10 23 11 24 15 16 2 8 25 14 16 2 8 24 20 11 2 8 25 19 11 2 8 23 15 16 2 9 25 13 16 2 9 31 32 11 32 31 1 33 34 19 34 33 15 35 36 7 36 35 3 1 7 24 18 15 1 7 24 20 13 1 7 25 13 19 1 7 25 14 18 1 7 25 18 14 1 7 25 19 13 19 13 5 6 22 19 15 3 6 22 15 18 4 6 22 15 19 3 6 22 20 13 4 6 22 20 14 3 6 22 10 21 17 14 3 8 21 17 14 5 9 21 12 20 3 8 21 12 20 4 9 21 17 15 3 8 21 17 15 4 12 4 8 25 16 12 4 10 23 16 17 5 8 24 11 17 5 9 23 11 12 5 8 24 16 12 5 9 23 16 23 20 11 2 9 25 18 11 2 9 23 14 16 2 10 24 13 16 2 10 23 19 11 2 10 24 18 11 2 10
Notice that in this figure, each upper case letter and each lower case letter appears once in each row, column and diagonal indicating that this is a cyclical magic square. The fact that each letter appears once in each broken diagonal indicates that it is pandiagonal. The values for the capitol letters may be assigned the values 0, 5, 10, 15 and 20 (in any order). For the lower case letters, the values are 1, 2, 3, 4, 5, again in any order. Or the Capitol letters may represent the low numbers with the high numbers assigned to the lower case letters. Now, there are 10 ways we can select A. It may be any one of the set of 5 high numbers or any one of the set of five low numbers. For each of these 10 ways there are 4 ways to select B (any of the four remaining numbers from the set A was chosen from. For each of these 40 ways, there are 3 ways to select C. And finally two ways to select D (and 1 way only to select E). Similarly, for each of these 240 ways, there are 5 x 4 x 3 x 2 x 1 ways to select the lower case values. This gives us a total of 240 x 120 = 28,800 = 2(5!)^{2}. Dividing this figure by 8 gives us the 3600 basic order 5 pandiagonal magic squares. The complete list of 3600 order5 pandiagonal magic squares are available for download. Cyclical pandiagonal magic squares Here I will present a condensation of what we have been discussing and a comparison with orders 7 and 11 cyclical pandiagonal magic squares. Order5 pandiagonal magic squares are of a type called Cyclical or Regular. This type of magic square has each cell containing a number represented by a capitol and lower case letter, and each letter appearing once in each row and column. All orders 4 and 5 pandiagonal magic squares are of this type. Order7 contains 38,102,400 cyclical pandiagonal magic squares. (It contains 640,120,320 noncyclical (irregular) pandiagonal magic squares {Prof. Albert L. Candy}.) Order7 pandiagonal magic squares, have two possible noncyclical transformations. Namely 1357246 and 1473625. For a given order, n, (where n is a prime number) there are (n3)(n4)(n!)^{2}/8 basic cyclical pandiagonal magic squares.
Column [6] figures are obtained by:
