Still More Magic Squares

Barnard uses the terms bordered and inlaid and clearly defines the difference between the two types of magic squares. If you remove the green cells of this order-10 simple magic square and close up the spaces, you end up with an order-8 magic square. In both cases, the central order-4 square is pandiagonal magic. Remove the rose colored cells from this order-14 magic square and close up the array to obtain an order12 magic square. If the 5 green 4x4 squares are then removed, you obtain an order-8 magic square. Now remove all cells except those within the four 2x2 blue borders, and close up this array to obtain an order-4 pandiagonal magic square. Finally, combine all the original 2x2 bordered arrays from the original order-14 square to obtain an order-6 magic square. The squares resulting from the changes in these two squares may be seen here. Some of John Hendricks inlaid magic squares may be seen here.

[1] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270 

Collison Patchwork Squares

David M. Collison (U.S.A. 1933-1991) investigated generalized parts, from which he constructed what he called patchwork magic squares. [1] Here I show an order-14 magic square. He constructed it from four 4 x 4 elbows, four 6 x 4 tees, one 6 x 6 cross, and four 4 x 4 squares. His square used associated order-4 magic squares which I transposed in this figure, to pandiagonal magic, by moving two bottom rows to the top, and then two left columns to the right.  All figures sum to a constant in proportion to the number of cells in a line. S 14 = 1379 so S2 = 197, S4 = 394, and S6 = 591. Because all figures are magic, they can be re-arranged, using rotations and reflections to produce a large variety of different order-14 magic squares. [1] John R.  Hendricks, The Magic Square Course, self-published, 1991, 554 pages (pp 309-318) Another patchwork square by Collison is this order-7 overlapping square shown in Decimal and Radix-7 form. The Decimal normal, simple magic square with S = 175 contains an order-3 semi-magic square with S = 75, two 2 x 4 magic rectangles with S = 50 and 100, and an order-5 pandiagonal magic square with S = 125. The Radix-7 square is the equivalent of the above, but numbered from 00 to 66 (instead of 01 to 100). S = 330. The 3 x 3 semi-magic square contains all 2-digit combinations of  0, 3, and 6. S = 132. The two 2 x 4 rectangles have S = 66 and 165. The pandiagonal 5 x 5 sums to 231.

Gnomon Magic Squares

Gnomon: the remainder of a parallelogram after the removal of a similar parallelogram containing one of its
corners.-Merriam-Webster Online

A 3 x 3 array may be dissected into a 2 x 2 array and a 5 cell gnomon in 4 ways.
When the sum of the cells in the four 2 x 2 arrays sum to the same value, the large array is said to be a gnomon-magic square. [1]
A. is such a square. Each of the corner 2x2 regions sum to 20. Refer to these regions as C.
B. This gnomon square consists of the 9 smallest prime numbers. C = 33. It is possible to construct an infinite variety of other prime magic gnomon squares
with 2 in the center cell.
The smallest prime number C of any gnomon square with 9 distinct primes has C = 43. Can you construct it?
C.D.E.F. Here are four gnomon-antimagic squares. The four C's (of each) = 19, 21, 23, and 25.

G. The Lho-shu has a special significance in ancient Arab culture when it is considered in this psuedo-gnomon manner. [2]
The Muslim alchemist, Jabir ibn Hayyan, considered the entire material world based on 17.
He noted too, the series 1:3:5:8 formed the foundation for all other numbers.
28 is the typical moon number. It is also the number of letters in the Arabic (abjad) alphabet.
The 2x2 shown in green = 17, the gnomon = 28.
H. Any 3x3 magic square may be converted to a gnomon square by rotating the perimeter numbers 1 position. C = 20.
J. is a magic square of palindromes, with S = 666. It is an antimagic gnomon because the C’s = 989, 878, 898, and 787.
K. Rotating the outer numbers results in a gnomon square with C = 888.
L. The Durer (1514) magic square is an example of an order-4 Gnomon-magic square because the four cells in each quadrant sum to 34.This is true for all
432 order-4 squares of Groups I to VI-A.

[1] Charles W. Trigg, Third Order Gnomon-Magic Squares, Journal of Recreational Mathematics, 18:1, 1985-86, pp 25-36
    Charles W. Trigg, Third Order Gnomon-Magic Squares with Distinct Primes, Journal of Recreational Mathematics, 19:1, 1987, pp 1-2
    Charles W. Trigg, A Remarkable Group of Gnomon-Antimagic Squares, Journal of Recreational Mathematics, 19:3, 1987, pp 164-168
[2] Annemarie Schimmel, The Mystery of Numbers, Oxford University Press,1991, 0-19-508919-7

Pickover's Puzzle

On the cover of the now defunct REC [1], Clifford Pickover posed as a puzzle, an unusual number square.
You have 25 squares in a 5x5 array. They contain 12 ones, a single two, and the rest zeros. Rearrange the digits so the sum of each row, column, and the two
main diagonals are positive and even. See a. for the puzzle and b. for Pickover’s solution.

The following issue of REC contained a number of additional solutions, including c. from Dennis Sulllivan and a similar symmetrical one d. from Harry J. Smith.
An example of an unsymmetrical one is e., from Don Kirk, who reported that there is at least one solution with the 2 in an given position! Dr. Robert Stong
reported that this problem could be solved by starting with an order 5 magic square. First replace every odd entry with a zero, then every even entry with a one.
Finally, replace one of the zeros with a two. 

[1] Recreational & Educational Computing, Dr. Michael Ecker, Volume 12, Number 3 (and 4) January (and March) 1998. 

Knecht's Topographical Models

Imagine a magic square where each cell is a square tower equal to the height of the cell’s numerical value. Craig Knecht refers to this model as a topographical
magic square. He has explored this type of magic square for the feature of how much water would be retained in the central part of the square. Topographical
squares and some results of his discoveries are displayed on his new web site at http://www.knechtmagicsquare.paulscomputing.com/     

Figure A. shows top and side views of a 3 x 3 model (not magic) where water could be poured into the center cell which would retain 1 unit before pouring over the cell with height equal to 1. Of the 880 order-4 magic squares he found that 137 would retain no water. B. shows an order-4 that retains 7 units of water. Craig shows topographical squares of  other orders on his site.

On July 30, 2008,  I received this more complex topographical magic square from him. C. contains a 'lake' with an 'island' in it.

On his site Craig also demonstrates mass in regards to magic squares and some work with paths.
He also has a page about the Formula One constraint Logic programming language.
Craig Knecht, Walter Trump, Francis Gaspalou, and Peter Loly investigated moment of inertia of magic squares.

See updated information on topographical magic squares on my 2009 and 2010 Update pages.

Steimel's Missing Digit Squares

Wilfred Steimel of Germany specializes in unusual magic squares. Here are two of his non-normal magic squares in which some digits are not used. I received these by email on May 31, 2008. a. is pandiagonal magic but contains no 0. 6, 7, 8, or 9 and so is not normal. S = 165 b. is non-normal, simple, concentric, and contains no digit 9. S for orders 3, 5, 7 and 9 are 183, 305, 427, and 549. The sum of the 9 colored cells in each of the 3 sets equals the S9 value, 549.

Boyer's Multimagic

Recently Christian Boyer added a new Update page to his excellent Multimagic site. I leave it to you to explore these new additions. Here I show two representative squares from elsewhere on his site.  Since 2007, Lee Morgenstern has constructed several orders 4, 5, and 6 semi-magic add-multiply magic squares, of which this 6x6 one is perhaps the best. He proved that orders 3 and 4 add-multiply magic squares are impossible. Is an order 5, 6, or 7 possible? There are no known add-multiply magic squares smaller then orders 8 and 9. The first orders 8 and 9 were constructed by W. W. Horner in 1955 and 1952 respectively. S = 360   P = 8491392000 All rows, columns and the 2 diagonals sum correctly All rows and columns produce the same product, but the 2 diagonals are incorrect To the right is a 6x6 magic square of cubes constructed in 2008  by Lee Morgenstern. It uses all the integers from -18 to 18 (except 0). S = 0. It is unknown if regular magic squares of cubes are possible for orders 4-8 (order-3 has been proved impossible). Christian Boyer constructed the first known orders 9, 10 and 11 normal magic squares of cubes in late 2006.

Add-Multiply Square Square of Cubes

Gutierrez's Semi-associated

Edward Gutierrez has a new site with the long name ODD MAGIC SQUARES USING TEN NEW ALGORITHMS AND PARITY THE WHEEL, LOUBÈRE AND MÉZIRIAC METHODS

This site introduces two new methods used for the construction of semi-associated odd magic squares. The method consists of pairing numbers in complementary
fashion, partitioning these complementary pairs into groups and placing these groups into a square in a certain order using parity as an aid.

This example is taken from his New Modified Wheel and De La Loubère Methods
Some of the complementary pairs are symmetrical around the center cell, but some pairs are not.

Loly's Gray Code Square

In one of his recent papers [1] Peter Loly [2] discusses a large variety of magic square and hypercube issues. Included is mention of an unusual pandiagonal
non-magic number square. If the decimal integers (0-15) are converted to a special binary number system called Gray code. binary, adjacent cell numbers, in
both the horizontal and vertical direction change by just 1 digit. This example is one of 48 such squares out of a total of 3,465,216 order-4 pandiagonal non-magic
squares.

Gray code Definition: An ordering of 2n binary numbers such that only one bit changes from one entry to the next.
An n-bit Gray code corresponds to a Hamiltonian cycle on an n-dimensional hypercube.
Hamiltonian cycle Definition: A path through a graph that starts and ends at the same vertex and includes every
other vertex exactly once.

The gray code is sometimes referred to as reflected binary, because the first eight values compare with those of the last 8 values (of regular binary), but in reverse
order.

This table is presented for the convenience of those not too familiar with the binary (radix 2) number system, or with the Gray code.

[1] Peter D. Loly, Franklin Squares: A Chapter in the Scientific Studies of Magical Squares, Complex Systems, 17 (2007) 143–161
[2] Some of his other papers on magic squares are at http://home.cc.umanitoba.ca/~loly/

Gaspalou Enumerations

Has a site at http://www.gaspalou.fr/magic-squares/ with much material on the methods and tools for enumeration of magic squares.

Among other results he confirmed the total (275,305,224) for order-5 squares. This value was first determined by Richard Schroeppel in 1973, then independently
confirmed by Walter Trump.

Campbell's Pan anti-magic squares

Neil Campbell has discovered a new class of anti-magic squares with features similar to the pandiagonal magic square.

A normal anti-magic square has sums for the orthogonal lines and the main diagonals that must be unique and consecutive. (if the sums are not consecutive,
the figure is a heterosquare.)

By changing the above definition slightly, it is possible to construct a square with the above features plus all the broken diagonals also summing correctly.
This means that the square can then by transformed to another pandiagonal (nasik) anti-magic square by moving a line from one side to the opposite side.

The slight change in the definition is: A pan-anti-magic square must contain all numbers from 0 to m2 and it must have pan-sums that can be mapped to a set bounded by the magic constant plus or minus half the set size. The set must have all consecutive numbers except skipping the magic sum. The order-4 example has 32 sums in consecutive order. Sums range from 14 to 46 skipping the magic constant 30. I have a page on anti-magic and heteromagic squares here.

Neil's father Dwane has a web site that discusses Nasik (Perfect) magic hypercubes and has some hypercube generator programs available for downloading.
He discusses at some length, theory of construction and methods of testing nasik hypercubes. His site is at http://home.earthlink.net/~dwanecampbel/.

Dürer Magic Square

Albrecht Durer (1471-1528 was a skilled artist and engraver in Germany. On of his most famous works, "Melencolia" was produced in 1514.  A prominent feature of this work was an order 4 magic square. This square is notable for it’s early date of construction, but also because it has a great many features aside from the standard rows, columns and main diagonals summing correctly. It is 1 of the 48 associated semi-pandiagonal order 4 squares of Dudeney’s Group III. All have the same features. There are a total of 880 distinct magic squares of order-4. All were enumerated by Bernard Frénicle de Bessy (1605-1675) and published (posthumously) in 1693. [1]

I recently ran across a paper by William H. Press, Did Durer Intentionally Show Only His Second-Best Magic Square? [2] It shows a variation.
By simply exchanging the second and third rows the the magnitude of the numbers in the four columns  alternately descend and ascend.

Patterns that sum to 34. The 5 rows, 5 columns, 2 main diagonals, 2 pairs of broken diagonals plus the 8 other patterns.

16 3 2 13 9 6 7 12 5 10 11 8 4 15 14 1 W. Press alternate square

This section was written and posted on March 10, 2012 when it was brought to my attention that my website made very little mention of this most famous square.
 Thanks, Paul Michelet.

[1] http://magic-squares.net/order4list.htm
[2] www.nr.com/whp/notes/DurerSquare.pdf 

Magic squares are distinguished into two classes, ordinary and perfect,...'
If the sums of the numbers in the broken diagonals, equal in every case, the
sums found in the vertical and horizontal rows, the square is perfect.[1]

Emory McClintock, On the 2^nd page of his paper Most Perfect Forms of Magic Squares, he states

By reason of these remarkable properties pandiagonal squares have received from
some writers, beginning with La Hire, the name ‘perfect”, and with others,
beginning with Lucas, the name “diabolic”.[2]

The term Nasik, we know, was introduced by A. H. Frost in 1866 [3] for pandiagonal magic squares and pandiagonal-like magic cubes. It was refined by
C. Planck in 1905 [4][5] to cover all dimensions of hypercubes where all possible lines sum correctly!

See my Nasik article for reasons why we should replace the term perfect with the term nasik!

[1] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270
[2] Emory McClintock, On the Most Perfect Forms of Magic Squares, with Methods for their Production, American Journal of Mathematics, 1897, 19, pp 99-120.
[3] A. H. Frost, Invention of Magic Cubes, Quarterly Journal of Mathematics, 7, 1866, pages 92-102. See page 99, para. 23 and page 100, para. 26.
[4] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905 (Available from The University Library, Cambridge).
[5] W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages 365,366, by Dr. C. Planck.
Re-published by Dover Publ., 1960 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373

	This page was originally posted August 2008 It was last updated March 09, 2012 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz