Magic Square Update2009

Dr. Robert Dickter (DDS) announced in an email of Aug. 18, 2009 a series of magic squares with identical features to the order3. He referred to them as Luo Shu format squares and shows a number of examples on his new website at www.luoshu.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, 2^{nd } 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 3^{rd} member of the triad (y1) is adjacent and above 'x'. The product of x and y equals the constant of the magic square, while x^{2} 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. Beginning with
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
magic squares.
[1]
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 [1] or [2] 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 order7 with 1 lake containing 365 units, and another order7 with a lake and a pond that contains a total of 378 units. Are these the maximums for order7 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 order7 number square with numbers 1 to m^{2}. [4] [1] See
Still More Magic
Squares In an email dated August 12, 2009 Francis Gaspalou announced that he had successfully enumerated the number of order6 selfsimilar 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 selfsimilar basic magic squares of order6 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/magicsquares/. [1] Most of Matsumi
Suzuki’s site is now at
http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.total.html
How many Bordered order 6 In an email March 29, 2009, Harry White sent a list of 140 borders to place around a nonnormal order4 magic square consisting of consecutive numbers. His web site [1] on this subject includes scripts to construct bordered squares of various orders.
In the illustration, A is a bordered order6 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!
Totals for
some distinct orders of bordered magic squares are: For concentric order6 magic squares Harry found the total was 736,347,893,760. This number was confirmed by Francis Gaspalou in July 2009. [1] http://users.eastlink.ca/~sharrywhite/BorderedMagicSquares.html Totally Irregular Magic Squares
On August
24, 2009, Francis Gaspalou introduced a new type of magic square which he
calls totally irregular.
A. (above)
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 [2] 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 order6, 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 order7. [3] 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. [4] 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. [1] Francis
Gaspalou’s web site is http://www.gaspalou.fr/magicsquares/. Postage Stamps and Magic Squares
After retiring from the
Department of Mathematics and Statistics, McGill University, in 2005,
Professor Emeritus George Styan
[1]
became interested again in magic squares. From the abstract of Some Comments On Philatelic Latin Squares From Pakistan [2] 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
(18811938), founder of the Republic of Turkey as well as its first
President, and QuaideAzam Muhammad Ali Jinnah (18761948), 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.
[1] George P. H. Styan may be reached at
styanatmath.mcgill.ca
On September
6, 2009 I received an email from Lee Sallows with two documents attached
[1][2]. Folowing are the first two paragraphs of NEW ADVANCES WITH 4 X 4 MAGIC SQUARES by Lee Sallows Introduction
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 nonnormal 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 lee.sal@inter.nl.net. 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.
Here are
3 example order4 squares taken from his index ordered list of 1040 magic
squares using the above number set (figures A, B, C).[3]
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 NONARITHMETIC progression of 16 integers that yield 880 4x4 magic squares.[2] 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 [4] 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 nonconsecutive numbers. Of course, from Lee Sallows and most other mathematicians perspective, the normal magic squares are only a subset of all number magic squares!
[1]
New Advances With 4 X 4 Magic Squares
by Lee Sallows New_Advances_(4).doc That Amazing 1089 and the Lho Shu The 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? [1]
[1] Adapted from Emanuel Emanouilidis: Journal of Recreational Mathematics:29:3:1988:177178
On August
18, 2009, Peter Loly reported (by email) a minor error in W. S. Andrews
Magic Squares and Cubes.
[1] Dr. Crypton, Timid Virgins Make Dull Company, Penguin Books, 1985, p 149, 0140080430
I received
these 4 related magic squares from Ed Shineman Jr. of New York on January
9, 2001
The magic constant of the 9 order3 squares form another order3 square.
Other Shineman creations on my site are on my Unusual Magic Squares and my Material from REC pages.
