Magic Square Update-2009
Dr. Robert Dickter (DDS) announced in an email of Aug. 18, 2009 a series of magic squares with identical features to the order-3. He referred to them as Luo Shu format squares and shows a number of examples on his new website at www.luo-shu.com
Here I show the
Luo Shu and 3 examples followed by their common features. These magic squares
are easy to construct using a modified de la Loubére method.
Some relationships between key numbers in all Luo Shu format magic squares.
‘a’ and ‘b’ are two consecutive numbers whose sum equals the order of the square. 'a' always appears in the bottom row, 2nd cell from the right. 'b' always appears in the top right hand corner.
'y' is always the center cell. 'x' is adjacent to the left of 'y', and the 3rd member of the triad (y-1) is adjacent and above 'x'. The product of x and y equals the constant of the magic square, while x2 times y gives the total of all integers in the square. These 3 cells always form a pythagorean triad in a Luo Shu format square. They are indicated in red in the illustrations.
order 9, the pythagorean triplet associated with the next higher order also
appears in the square in the position shown with green numbers.
In August, 2008, I posted an article about Craig Knecht’s Topographical
The basic premise is that the ‘height’ of each cell is based on the value of the integer in that cell. Then cells that are lower then the surrounding cells may contain ‘water’. Please refer to  or  for a complete description of topographical magic squares.
I suppose, to be consistent, we must say that figure 1(A) contains 2 ‘lakes’ of 2 cells each, and 1 ‘pond’ and (C) contains 1 lake and 1 pond. This follows from the fact that the basic definition states that no water can flow diagonally between cells.
Walter Trump has found an order-7 with 1 lake containing 365 units, and another order-7 with a lake and a pond that contains a total of 378 units. Are these the maximums for order-7 magic squares?
E. This is a number (not magic) square. (Disregard the fainter numbers. They are just inserted to complete the square.) The lake and the 2 ponds together retain 488 units of water. This is the maximum possible for an order-7 number square with numbers 1 to m2. 
Still More Magic
In an email dated August 12, 2009 Francis Gaspalou announced that he had successfully enumerated the number of order-6 self-similar magic squares (see example). He reported that there were 67,704,146,804,736 different squares of type a1+f1 = 37 (when counting all of the squares). There are the same number when a1+a6=37 so the total number of self-similar basic magic squares of order-6 is exactly 16,926,036,701,184 (ie 2 times 67,704,146,804,736/8).
Francis has a site where he shows results of various magic square enumerations and methods at http://www.gaspalou.fr/magic-squares/.
 Most of Matsumi
Suzuki’s site is now at
How many Bordered order 6
In an email March 29, 2009, Harry White sent a list of 140 borders to place around a non-normal order-4 magic square consisting of consecutive numbers. His web site  on this subject includes scripts to construct bordered squares of various orders.
In the illustration, A is a bordered order-6 magic square. B. is a concentric magic square, but in this special case, the border consists of consecutive numbers. This is not required, and is in fact, quite rare!
some distinct orders of bordered magic squares are:
For concentric order-6 magic squares Harry found the total was 736,347,893,760. This number was confirmed by Francis Gaspalou in July 2009.
24, 2009, Francis Gaspalou introduced a new type of magic square which he
calls totally irregular.
is the totally irregular simple magic square received from Francis
Gaspalou. Aug. 24, 2009
Magic squares may be constructed by using two subsidiary squares which are then added on a cell by cell basis to form the final square. Such squares may then be divided into 2 broad classifications.
A Regular magic square  is one where each number (in the classical version, each letter) appears once in each row, column, and diagonal. If such a square could exist for order-6, then each line would total 21 in subsidiary square A and 90 in subsidiary square B. Prof. Candy reported that there are 38,102,400 regular magic squares of order-7. 
An Irregular magic square is one where this is not the case and a number may appear more then once (or not at all) in at least 1 row, column, or diagonal. For example, there are no irregular pandiagonal magic squares of order less the 7, but both Trump and Gaspalou found that there are many more then the 640,120,320 different ones of that order reported by Prof. Candy. 
A Totally Irregular magic square (as the one shown above) is one where all the lines are irregular in both subsidiary squares. The total number of these (for any order) is still unknown.
Gaspalou’s web site is http://www.gaspalou.fr/magic-squares/.
After retiring from the
Department of Mathematics and Statistics, McGill University, in 2005,
Professor Emeritus George Styan
became interested again in magic squares.
From the abstract of Some Comments On Philatelic Latin Squares From Pakistan 
We explore the use of Latin squares in printing postage stamps, with special emphasis on stamps from Pakistan. We note that Pakistan may be the only country to have issued postage stamps in 2x2; 3x3; 4x4 and 5x5 Latin square formats: we call such sets of stamps philatelic Latin squares (PLS)….
Here are three illustrations from that paper.
The sheetlets displayed
in Figures 1.1 and 1.3 are examples of 2x2 philatelic Latin squares (PLS).
The stamps in the left panel (fig. 1.1) depict Mustafa Kemal Atatürk
(1881--1938), founder of the Republic of Turkey as well as its first
President, and Quaid-e-Azam Muhammad Ali Jinnah (1876--1948), who is
generally regarded as the founder of Pakistan.
The authors do not
discuss magic squares per se, but consider various aspects of Latin
Squares and their newly defined PLS. They also discuss Gerechte Latin
squares and the somewhat related Sudoku square in some depth.
 George P. H. Styan may be reached at
6, 2009 I received an email from Lee Sallows with two documents attached
Folowing are the first two paragraphs of NEW ADVANCES WITH 4 X 4 MAGIC SQUARES by Lee Sallows
One of the best known results in the magic square canon is Bernard Frénicle de Bessy's enumeration of the 880 ‘normal’ 4´4 squares that can be formed using the arithmetic progression 1,2,..,16. A natural question this suggests concerns non-normal squares: Is 880 the largest total attainable if any 16 distinct numbers are allowed?
The answer is no. A computer program that will generate every square constructable from any given set of integers has identified 1040 distinct squares using the almost arithmetic progression: 1, 2, 3, 4, 5, 6, 7, 8,10,11,12,13,14,15,16,17. Note the doubled step from 8 to 10. Extensive trials with alternative sets make it virtually certain that 1040 is the maximum attainable (or 8 ´ 1040 = 8320, if rotations and reflections are included), although an analytic proof of this assumption is lacking. A listing of the 1040 squares can be had on request via email at email@example.com.
In this paper he goes on to examine the FERTILITY of alternative sets of 16 numbers, meaning the number of magic squares yielded by each. Tables listing the fertilities of both symmetric and asymmetric sets are given.
3 example order-4 squares taken from his index ordered list of 1040 magic
squares using the above number set (figures A, B, C).
The quantity of squares in each of the 12 (traditional) Dudeney groups for this number set is
The second document I received was A SET USING 1, 2, 3, 4, 6, 7, 8, 9, 10 11, 12, 13, 15, 16, 17, 18.
The second document received was a NON-ARITHMETIC progression of 16 integers that yield 880 4x4 magic squares.
This list of 880 squares (using 8 complementary integer pairs) includes 6 with distributions that are DIFFERENT to the usual 12 Dudeney types, such as the examples D and E above.
If you wish, you may review the 12 original Dudeney groups here.
A subsequent paper  received on 9/9/09 from Lee Sallows identifies a total of 22 additional Dudeney type diagrams and a representative magic square for each. Unlike the original Dudeney types, these apply only to magic squares that consist of non-consecutive numbers. Of course, from Lee Sallows and most other mathematicians perspective, the normal magic squares are only a sub-set of all number magic squares!
New Advances With 4 X 4 Magic Squares
by Lee Sallows New_Advances_(4).doc
That Amazing 1089 and the Lho ShuThe Luo Shu multiplied by 1089 gives us a magic square with the magic sum of 16335.
The 1 MSD gives us the original square, and the 1 LSD give us a 180 degree rotation of the original magic square.
Oct. 20/12 Thanks to Lorenzo Susican Jr. of The Phillppines for spotting the error in my magic square sum (I had 16385).
Eight additional squares derived from square A. How many more variations can you find? 
 Adapted from Emanuel Emanouilidis: Journal of Recreational Mathematics:29:3:1988:177-178
18, 2009, Peter Loly reported (by email) a minor error in W. S. Andrews
Magic Squares and Cubes.
 Dr. Crypton, Timid Virgins Make Dull Company, Penguin Books, 1985, p 149, 01400-80430
these 4 related magic squares from Ed Shineman Jr. of New York on January
The magic constant of the 9 order-3 squares form another order-3 square.