The example on this page is used to show some of
the number patterns that sum to the magic constant.
Also some definitions to help you understand my explanation of this square.
See how five numbers can combine in 1128
(?)
ways to sum to 65.
The Order5 Magic
Square
This
order5 pandiagonal, associative, complete, and selfsimilar magic
square has the following combinations of 5 cells summing to 65:
Total
combinations of five cell patterns summing to 65 = 1128
Do you arrive at the same total?
(Note: see addendum for correct total)

1 
15 
24 
8 
17 
23 
7 
16 
5 
14 
20 
4 
13 
22 
6 
12 
21 
10 
19 
3 
9 
18 
2 
11 
25 

Combination 
# 
Example 

Wrap
Example 
Rows 
5 
1, 15, 24, 8, 17 
wraparound is not counted 

columns 
5 
1, 23, 20, 12, 9 
wraparound is not counted 

diagonals 
10 
1, 7, 13, 19, 25 
includes wraparound 
15, 16, 22, 3, 9 
corners of 3 x 3 squares + center 
25 
1, 24, 7, 20, 13 
including wraparound 
12, 10, 18, 1, 24 
corners of 5 x 5 squares + center 
25 
1, 17, 13, 9, 25 
including wraparound 
23, 14, 10, 1, 17 
corners of 2 x 2 rhombics + center 
25 
23, 15, 7, 4, 16 
including wraparound 
11, 3, 25, 17, 9 
corners of 3 x 3 rhombics + center 
25 
20, 24, 13, 2, 6 
including wraparound 
2, 6, 25, 14, 18 
corners of 4 x 4 rhombics + center 
25 

all are wraparound 
1, 22, 8, 19, 15 
corners of 5 x 5 rhombics + center 
25 

all are wraparound 
7, 20, 23, 1, 14 
Plus 
58 
combinations of any 2 symmetrical cell
pairs plus the center cell. (see diagrams below) 
1, 15, 13, 11, 25 wraparound doesn’t work 
Plus 
900 
combinations of 8 nonsymmetrical patterns
(see diagrams below) 
1, 8, 17, 14, 25 15, 17, 1, 23, 9 
July 31/99
Not so! 
See the addendum now at the end of
this page, which explains a very big error in my count!
I have chosen to leave the original page as is to better explain how
wraparound can cause apparently different patterns.
Nov. 29, 2012 Another Addendum
This one, from Oscar Lanzi, shows the actual count is only 120 magic
patterns in all pandiagonal squares, although there are more
in some. 
2
Definitions
Pandiagonal
Also known as Diabolic, Nasic,
Continuous, Indian, Jaina or Perfect M.S. To be
pandiagonal, the broken diagonal pairs must also sum to the
magic constant. This is considered the top class of magic
squares. Some pandiagonal magic squares are also
associative (order 5 & higher) . Because of the vast number of
combinations possible, individual magic squares may contain other
unique features that make them more magic.
There is only 1 basic order 3 magic square and it is not
pandiagonal.
Of the 880 basic order 4 magic squares, only 48 are pandiagonal
and none of these are associative.
Order 5 has 3600 basic pandiagonal magic squares (Only 36
essentially different).
Order 7 has 678,222,720 basic pandiagonal magic squares. (38,102,400
regular plus 640,120,320 irregular panmagic squares). (Order 8 has
more then 6.5 billion pandiagonal magic squares.)
See Benson & Jacoby, New
Recreations With Magic Squares, Dover Publ., 1976. 
1 
15 
24 
8 
17 
23 
7 
16 
5 
14 
20 
4 
13 
22 
6 
12 
21 
10 
19 
3 
9 
18 
2 
11 
25 
copy of
above order 5

Associative
A magic square where all pairs of cells diametrically equidistant
from the center of the square equal the sum of the first and last
terms of the series, or m^{2} + 1. Also called Symmetrical or
Regular. The center cell of odd order associated magic
squares is always equal to the middle number of the series. Therefore the
sum of each pair is equal to 2 times the center cell and the sum of any 2
symmetrical pairs plus the center cell is equal to the constant. This
permits a great many combinations (the order 5 square above has 58 of this
type).
In an even order magic square, the sum of any 2 symmetrical pairs
will equal the constant.
There are NO singlyeven Associated magic squares.
The one order3 magic square is associative.
Selfsimilar
means that when each number is changed to its complement (i.e.
subtracted from m+1), an identical magic square is formed
but rotated 180 degrees. This term was coined by Mr. Mutsumi Suzuki
who discovered six such squares. See his excellent site at http://www.pse.che.tohoku.ac.jp/~msuzuki/
(This is now available at
http://www.archive.org/ (the Wayback Machine)
Wraparound
means when you go off of one edge, continue (wraparound) to the
corresponding cell on the opposite edge. The easiest way to
accomplish this is to lay out the magic square in a repeating
pattern on the plane as shown to the right. Then any 5 x 5 array
is an equivalent pandiagonal magic square. So, to complete a
line, you can just keep going in the same direction. The red cell
values illustrate wraparound when applied to a broken diagonal
pair. Wraparound works only with pandiagonal magic squares. 

Five
Symmetrical cell
pairs
A Symmetrical cell pair is defined as 2 cells that are
symmetrical around the center cell i.e. 1 & 25, 8 & 18, 4 & 22, 7 & 19
etc. Wraparound doesn't work with these. The 3 squares illustrate the 5
basic pairs. Each of these may be rotated and/or reflected and used in
any unique combination of 2 pairs plus the center cell.

Note that some of these combinations
are duplicates of the basic patterns counted at the start. Example;
if the 2 patterns in the center diagram are lined up, this
combination constitutes a main diagonal which has already been
counted. 
Eight
Nonsymmetrical patterns
The seven patterns on the left (below) each appear 25 times because of
wrap around and 4 times due to rotation ( i.e. 7 x 25 x 4 = 700).
They are symmetrical about a horizontal, vertical or diagonal axis and so
cannot be reflected for unique solutions.
That pattern 1 will always be magic in an order5 pandiagonal magic
square, if it is rotated so the 3 adjacent cells are in any of the 4
corners was proven by Rosser and Walker in 1939.
[1]
The pattern on the right is not symmetrical around a vertical,
horizontal, or diagonal axis, so appears 25 times because of wrap around,
4 times due to rotation and 2 times because of reflection (i.e. 25 x 4 x 2
= 200).
Starting a given pattern on a different cell, in combination with a
rotation or a reflection, may result in the same 5 cells being used.
Example, start the first pattern with 9, 23, 1, 15, 17. Now rotate the
pattern 90 degrees clockwise and start with 15,17, 1, 23, 9. Both
combinations use the same 5 cells, but in a different arrangement. I
consider them different combinations.
Of course, we could say the same about the use of wraparound when
computing the number of combinations for rows, columns and the diagonals.
However, by convention, these extra combinations are not counted.
[1] R. Rosser and R.J Walker, The Algebraic Theory of
Diabolic Magic Squares, Duke Mathematical Journal, Vol. 5, No, 4, Dec.
1939 pp 705728 (p.717)
Addendum1
On June 28/99, I received an email from Aale de Winkel
advising me that he thought nonsymmetric pattern 1, above, was actually
equivalent to the wraparound 2 by 2 rhombic, and patterns 2 and 8 were
equivalent to the wraparound 2 by 2 square.
Subsequent investigation on my part confirmed these equivalents,
and also that four of the other five patterns are also equivalent to symmetric
patterns already counted.
1. is equivalent to wraparound 2 by 2 rhombic ( now called
plusmagic  see Quadrant Magic Squares)
2. is equivalent to wraparound 2 by 2 square ( now called
crosmagic)
3. is equivalent to wraparound 3 by 3 rhombic corners.
4. is equivalent to wraparound 5 by 5 square corners.
5. is just a reflected 4. (another goof!).
6. is unique, so add 100 combinations to the total.
7. is equivalent to wraparound 5 by 5 square corners.
8. is equivalent to wraparound 2 by 2 square
The three diagrams below demonstrate the equivalence of the 2 x
2 rhombic (plusmagic) and nonsymmetric pattern 1.
Diagram A shows pattern 1 within the magic
square and the plusmagic with two cells outside of the square. diagrams
B and C show the patterns shifted down and to the
right. In each of these cases, the plusmagic pattern is within the square and
two cells of the pattern 1 outside.
Yet to be determined. Should the rotated
nonsymmetric patterns be counted and included in the total?
For now, let us say that there are 328 different combinations!
By the way. There are 1394 sets of 5 numbers, from the first 25 consecutive
numbers, that sum to 65!
Addendum2
How many magic sums are there in a general 5x5
pandiagonal magic square? It turns out that the answer is actually just 120, not
328 or any larger number.
All 5x5 pandiagonal magic squares are regular; they are sums of two pandiagonal
Latin squares. We may render these Latin squares with labels:
A1 A2 A3 A4 A5 B1 B2
B3 B4 B5
A3 A4 A5 A1 A2 B4 B5
B1 B2 B3
A5 A1 A2 A3 A4 B2 B3
B4 B5 B1
A2 A3 A4 A5 A1 B5 B1
B2 B3 B4
A4 A5 A1 A2 A3 B3 B4
B5 B1 B2
To identify a magic sum, we pick five positions in the first Latin square and
the corresponding positions in the second Latin square, requiring each of the
ten distinct labels (A1 through A5, B1 through B5) to appear exactly once in our
selected squares. It turns out that there are just 120 ways to do this. Five are
the rows, five are the columns, ten are the diagonals and the rest are
“quincunx” sums in the form of plussign or x patterns.
There are some 5x5 pandiagonal squares with more magic sums, such as associative
squares, because of additional symmetry. But these additional patterns are not
general to all 5x5 pandiagonal squares and are not preserved by the
transformations that preserve pandiagonality. For example, if we cyclically
permute the rows of a 5x5 associative square, we no longer have an associative
square. The extra sums in the associative square, beyond the 120 general ones
described above, are then lost. The 120 remain.
Oscar Lanzi November, 2012
