In addition to the two main diagonals, each diagonal that wraps around at the edges of the square, sums to the magic constant:
Squares that are pandiagonal and associative are called ultramagic. Two special classes of pandiagonal squares are:
where n is a doubly even order. See note.
Squares that are compact and complete are called most-perfect.
PandiagonalSquares makes these squares using the methods outlined below. MagicSquaresSODLS also makes pandiagonal and ultramagic squares for some orders.
Order5Special makes the 3600 order 5 pandiagonal magic squares.
The Siamese method. See "Demanding the square is panmagic" in math behind the Siamese method. This method does not work when the order is a multiple of 3.
Squares of order n = 3k, k = 3, 5, 7, ... can be made by Margossian's method. See under SYMMETRICAL AND PANDIAGONAL SQUARES, page 208, in MATHEMATICAL RECREATIONS AND ESSAYS.
These are compact_{n/3}.
In THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS, page 255, Clifford Pickover says "Until the 1990s, everyone believed that it was impossible to construct a regular pandiagonal order-9 magic square...".
Actually, these squares were known long before the 1990s. For example, below is one from 1908 and another from 1878.
This is the first of five given in A.Margossian, DE L'ORDONNANCE DES NOMBRE DANS LES CARRÉS MAGIQUES IMPAIRS, 1908, .txt:
And, this one is given in A.H.Frost, ON THE GENERAL PROPERTIES OF NASIK SQUARES, 1878, .txt:
Note: 72 at row 3, column 6 is corrected to 69.
This order 15 appears in the same article, .txt:
Margossian's method. These are most-perfect. Except for order 4, they are U zigzag_{n/4}.
Planck's "A-D method" from associative magic squares. The method is:
In the example below, (e E), (f F), ... represent complement pairs. The cells of one broken diagonal are shown in color.
The squares made by this method are complete.
There are 48 order 4 pandiagonal magic squares. These are TYPE I in the classification by Dudeney. They are all most-perfect.
There are no pandiagonal magic squares of singly-even order.
This was shown by A.H. Frost (1878) and C. Planck (1919). See
references.
There are singly-even magic squares in which only four diagonals do not sum to
the
magic constant.
This is the minimum because:
PandiagonalSquares makes these squares using a process involving three main steps:
where, for example, L' is the symmetric complement of L.
Possible patterns for n = 6, 10, 14, 18 are:
Orders 6, 14, 22, 30, ... and 10, 18, 26, 34, ... have similar patterns, (differences shown in blue).
Move 1 or more row(s) and/or column(s) from one side to the opposite side to make the square magic.
A pandiagonal magic square remains pandiagonally magic if a row or column is moved from one side of the square to the opposite side. So, each order n square makes a total of n^{2} squares in this way. A near-pandiagonal magic square can be transformed into (n-2)^{2} near-pandiagonal magic squares in this way, (and 4(n - 1) semi-magic squares).
In addition to the two main diagonals, all the diagonals, (parallel to only 1 main diagonal), that wrap around at the edges of the square, each add up to the magic constant:
There are 77580 order 5 pandiagonal 1-way magic squares.
Here are order 7 and order 9 pandiagonal 1-way magic squares.
There are no pandiagonal 1-way magic squares of singly-even order. See Planck's proof.
For doubly-even orders greater than 4, there are pandiagonal 1-way magic squares. Here are order 8, order 12 examples.
There are squares in which all diagonals have the magic sum but the rows and columns may not. One example of these is made by simply writing the ordered sequence 1 to n^{2}:
These are called natural squares. See Planck. See also reversible squares.
Ball, W.W.Rouse and H.S.M. Coxeter "MATHEMATICAL RECREATIONS AND ESSAYS,
Eleventh Edition"
https://epdf.pub/queue/mathematical-recreations-and-essays-11th-rev-ed.html, page 208
Candy, Albert L. "Pandiagonal Magic Squares of Prime Order"
http://babel.hathitrust.org/cgi/pt?id=uc1.b4250447;view=1up;seq=7
Candy, Albert L. "Pandiagonal Magic Squares of Composite Order"
http://babel.hathitrust.org/cgi/pt?id=mdp.39015010791260;view=1up;seq=5
Dudeney, Henry E. "Magic Square Problems"
https://archive.org/details/AmusementsInMathematicspdf
Frost, A.H. "ON THE GENERAL PROPERTIES OF NASIK SQUARES"
http://books.google.com/books?id=qxMLAAAAYAAJ&pg=PA34#v=onepage&q&f=false
Heinz, Harvey "Order 4 Magic Squares, Group I ...The pandiagonals"
www.magic-squares.net/order4list.htm#Group I
Heinz, Harvey "Ultra Magic squares - Walter Trump" http://www.magic-squares.net/trump-ultra.htm
Hospel, Ton "The math behind the Siamese method of generating magic squares"
http://www.xs4all.nl/~thospel/siamese.html
Margossian, A. "DE L'ORDONNANCE DES NOMBRE DANS LES CARRÉS MAGIQUES IMPAIRS"
https://archive.org/details/delordonnancede00marggoog
Pickover, Clifford A. "THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS"
https://epdf.pub/the-zen-of-magic-squares-circles-and-stars-an-exhibition-of-surprising-structure.html
Planck, C. "PANDIAGONAL MAGICS OF ORDERS 6 AND 10 WITH MINIMAL NUMBERS."
http://www.jstor.org/stable/27900742?seq=1#page_scan_tab_contents
Weisstein, Eric W. "Panmagic Square." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/PanmagicSquare.html