1. Science in the Universe of the Matrix Elements 1
Science in the Universe of the Matrix Elements 1..n2 Windsor 2007 June 1-3
Peter Loly & Ian Cameron
With Walter Trump, Adam Rogers & Daniel Schindel
Critical funding from the Winnipeg Foundation in 2003.
I am glad to have been invited by George Styan to talk to this group, and lead off a session on magic squares. This may be a pivotal event in what I like to call “The Science of Magical Squares”. Ian Cameron is my co-author for this talk and we both hope to begin fruitful discussions with many of you during the workshop. We are also to learn of your interests.

2. 35 48 3 1 6 40 42 19 34 28 21 20 46 7 11 26 38 13 45 33 9 18 36 27 25 23 14 32 41 17 5 37 12 24 39 43 4 30 29 22 16 31 8 10 44 49 47 2 15 15 12 21 10 7 2 6 17 18 22 25 23 13 3 1 4 8 9 20 24 19 16 5 14 11 8 1 6 3 5 7 4 9 2 64 9 17 40 32 41 49 8 2 55 47 26 34 23 15 58 3 54 46 27 35 22 14 59 61 12 20 37 29 44 52 5 60 13 21 36 28 45 53 4 6 51 43 30 38 19 11 62 7 50 42 31 39 18 10 63 57 16 24 33 25 48 56 1 2 9 4 29 36 31 34 32 30 7 5 3 6 1 8 33 28 35 20 27 22 11 18 13 25 23 21 16 14 12 24 19 26 15 10 17 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
1693 Frenicle counted 880 in order 4; 1972 Richard Schroeppel used a computer to obtain some 275 million in 5th order.

3. Overview
Focus on sequential integer square matrices with matrix elements 1..n2 (but make use of general properties of real square matrices).
Factor out the magic eigenvalue for semi-magic squares from characteristic polynomial, and thus for magic squares.
Singular value decomposition (SVD) analysis.
Compound squares – Kronecker product.
Maple, Mathematica, Scientific WorkPlace and MATLAB used where appropriate.

4. 12,544x12,544 compound magic square
Our largest n=12,544, ,351,936. Coloured on a rainbow scale. Memory: 620 Mb on a CD (IDL).
Reference: Chan and Loly, MTY

5. Pathway patterns 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
Patterns from cell to cell produce wonderful patterns.
Franklin?
Claude Bragdon

6. The Manitoba Magicians [with apologies to the International Brotherhood of Magicians (IBM), founded in Winnipeg in 1922]
John Hendricks, meteorologist (retired to Victoria, British Columbia)
Frank Hruska, Chemistry, University of Manitoba
Vaclav Linek and John Cormie (antimagic squares) University of Winnipeg
Peter Loly, retired December 2006, Senior Scholar, U Manitoba
Ian Cameron, Planetarium and Observatory supervisor, U Manitoba
Marcus Steeds, Frantic Films
Wayne Chan, Centre for Earth Observation Science (CEOS)
Adam Rogers, graduate student, genetic algorithms for astrophysics
Daniel Schindel, Michigan State, East Lansing, nuclear theory
Matthew Rempel, second degree in engineering
Russell Holmes, PhD Princeton
Gideon Garland, mass spectroscopy, Israel.
Red and Blue: undergraduates at U Manitoba; Red are published with PDL.

7. Modern Combinatorics Persis Diaconis (and company)
“Following MacMahon [45] and Stanley [52], what is referred to as magic squares in modern combinatorics are square matrices of order k, whose entries are nonnegative integers and whose rows and columns sum up to the same number j.”
11/PDF/v11i2r2.pdf
Counting integer points in polyhedral cones (de Loera, Beck, Ahmed, ...). Difficult to handle matrix elements 1..n2.

8. Doubly stochastic matrices
Shin, Guibas and Zhao,
CS dept., Stanford: footnote 5:
“The doubly-stochastic matrix is a N x N non-negative matrix, whose rows and columns sum to one.”

9. Normal (Classic) Magical Squares “magic” n-sum =n(n2 +1)/2
Semi-magic (SM)
All rows and columns
Magic squares (MS) include both diagonals
“not even semimagic”
Non-magic pandiagonal
No rows or columns
Example: serial squares of any order
Example: logic squares of orders n=2p
i.e., n=2,4,8,16,32
Antipodal constraint (local) associative or regular MS
Global constraints: pandiagonal MS, bent diagonal (Franklin), complete (McClintock), complement and pandiagonal.
LH: SM to Magic and varieties.
RH: non-semimagic.

10. Bagel torus topology rubber sheet geometry
Take a square sheet
Join a pair of opposite edges to form a cylinder
Bend the cylinder until its ends join
Also known as periodic boundary conditions.
Useful for thinking about pandiagonals.
Wrapping on a torus (bagel).

11. How Many Normal Squares?
1/8(n2)! distinct 1.. n2 squares
for n = 2 => 4.3.2/8 = 3 distinct squares
3 x 3: now = 45,360
4 x 4: 57,657,600 times 3-by-3 count
= 2,615,348,736,000 ( * 1012)
After listing the n =3 squares, we add constraints to reduce these numbers!

12. Three 2-by-2’s 1 2 3 4 1 2 4 3 1 3 4 2
S2 “Serial”, upper left, is pandiagonal and regular.
“Cyclic”, upper right, is affine.
“Scissors” lower left also affine.
Note L2=S2.

13. 4-by-4 serial S4 and logic L4 squares
Both are pandiagonal non-magic
Serial squares exist for any order
Logic squares or order 2p derive from Karnaugh maps and Gray code, e.g., edges: {0,1}; {00, 01,11,10} [Loly and Steeds, 2005] (incremented to 1.. n2)
Complementing alternate cells with 17, i.e., 17-x, yields a pandiagonal magic square (Meine and Schütt, Siemens) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 4 3 5 6 8 7 13 14 16 15 9 10 12 11
Logic square initially found from a matrix of personality types – Myers-Briggs (Carl Jung).

14. Serial Squares of order n characteristic polynomial xn-2(x2+ αnx+ βn )
EV’s βn
scaled 2
(5±√33)/2 -2 1 3
(15±3√33)/2
-18 9 4
17±3√41
-80 40 5
(65±5√209)/2
-250
125
Loly and Steeds, “A New Class of Pandiagonal Non-Magic Squares, Int. J. Math. Ed. Sci. Tech. 36 (2005) [IJMEST]
Sloane Integer Sequence A (unique from first 4 terms, checked 11) [
Walsh and Lehmann, “Counting rooted maps by genus. III: Non-separable maps, J. Comb. Th. Ser. B 18 (1975)
Need to complete last column and look for series; also for logic squares.

15. Serial and Logic squares: rank 2
SVD 2
S2, L2
½(√34±√26) 3 S3
½(√321±√249) 4
S4, L4
3√46±√334

16. Magic Square Counts: Trump (c. 2002)
semi-magic B:
normal magic squares C:
regular (associative) D:
pan-diagonal
E: ultramagic 3 9 1 4
68688
880 48 5
579,043,051,200
275,305,224
48544
3600 16 6
9.4597(13). 1022
(42). 1019 7
(17). 1038
(50). 1034
(51). 1018
1.21(12). 1017
20,190,684 8
1.0804(13). 1059
5.2210(70).1054
2.5228(14). 1027
>C8
4.677(17). 1015
Errors from repeated runs (hundreds or thousands?).

17. Backtracking
Schroeppel See Gardner, M., Mathematical Games, Scientific American (1976)
Pinn, K. and Wieczerkowski, C., 1998, Number of Magic Squares from Parallel Tempering Monte Carlo, International Journal of Modern Physics C, 9(4),
Trump, W. Notes on Magic Squares and Cubes,
Schindel, D. G., Rempel, M. and Loly, P., 2006, Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS, Proceedings of the Royal Society A: Physical, Mathematical and Engineering, 462,
(Screen savers)

18. Vector Spaces of Magical Squares
General 3x3 Lucas
General 4x4 Bergholt 1910
John Tromp & Peter Loly
- Haskell/Maple

19. Bergholt 1910 4-sum: A+B+C+D
A-a
C+a+c
B+b-c
D-b
D+a-d B C
A-a+d
C-b+d A D
B+b-d
B+b
D-a-c
A-b+c
C+a
Row.column, diagonals, corners, central quartet, red and purple

20. Rotation
A square has 8 phases obtained from any one by rotations and reflections.
It is convention to select one, but it turns out to be illuminating to study a second phase, either a 90° rotation or a flip.
This is illustrated next for the archetypal LoShu n = 3 magic square from China – probably 2 millennia old.

21. Lo-shu (A,B: Frank Hruska 1991) [SVD: 15, 4√3, 2√3 – Loly 2007]8 1 6
Top row
Det
EV’s 15 and
4,9,2
360
±2i√6
8,3,4
-360
±2√6
6,1,8 (A)
2,7,6
4,3,8
2,9,4
6,7,2
8,1,6 (B) 8 3 4 1 5 9 6 7 2

22. MATLAB’s magic(n)
Separate algorithms for producing one square in each case [Cleve Moler, MATLAB’s Magical Mystery Tour, Winter 1993, MathWorks Newsletter 7(1) 8-9]
Odd n – regular (associative) NONSINGULAR
Singly even – NOT regular, SINGULAR
Doubly even – regular, SINGULAR
1995 Kirkland and Neumann give EV’s and SVD formulae for n = 4k [Lin. Alg. and its Appls. 220: ].

23. Mattingly singular even order regular magic squares
1999 (preprint) Mattingly proof of singularity for even order [Am. Math. Monthly 107 (2000) ]
1999 Loly already had studied all singular squares in the 4th order set of 880 by Dudeney group, finding examples with just one non-zero eigenvalue.

24. 5th order regular magic squares
Mattingly had conjectured odd orders were non-singular.
2003 Schindel and Loly, using programming ideas from an n=6 hybrid backtracking code of Walter Trump, regenerated the 5th order set of some 275 million magic squares and found that of the 48,544 regular magic squares, 652 were singular with 2 zero EVs, AND four had 4 zero EVs.
The 16 ultramagic squares [Suzuki] are non-singular.

25. 7th order ultramagic squares Walter Trump
Trump found 20,190,684 of these squares, of which Schindel found 20,604 to be singular, with a pair of zero eigenvalues.
Trump also found that less than 0.06% of 7th order regular squares are singular.
Kerry Brock, “How many singular squares are there?” Math. Gaz. 89 (Nov. 2005)

26. W-42 Götz Trenkler “On the Moore-Penrose inverse of magic squares”
We leave this aspect of singular magic squares to GT.

27. Dudeney Groups
Henry Ernest Dudeney ( )

28. Number of ATA or AAT matrices
880 4th order magic squares
Dudeney groups
Number of each
SVD sets (63)
Number of ATA or AAT matrices
1,2,3
1. Pandiagonal,
2. Semi-pandiagonal
& semi-bent,
3. regular 48
3 (α,β,γ)
5 (2,2,1)
4,5,6A
Semi-pandiagonal 96 10
Not yet counted (NYC) 6B
Simple
208 26
NYC
7,8,9,10 56 22
11,12 8 2

29. Talks W-44 and W-08
Kimmo Vehkalahti will talk about the 640 singular magic squares in more detail than I can include in my talk.
La Lok Chu, in joint work with George Styan, will talk about various issues for the 880, including their work on odd powers of certain remarkable cases.
PDL will focus on a few examples and SVD.

30. SVD
Since we have found many magic squares with only a single non-zero eigenvalue, but rank 3 (or more),
SVD values give more information (akin to an X-ray).
Further, motivated by analytical results of Kirkland and Neumann, we turned to examining the eigenvalues of ATA, the (observability) Gramian matrix, a symmetric matrix, where the square root of its (positive) eigenvalues gives the SVD values, with the largest being the linesum eigenvalue.

31. all regular (group 3) ATA’s are bisymmetric (M=JMTJ) Example: F790 regular5 4 16 9 11 14 2 7 10 15 3 6 8 1 13 12
EV: 34,0,0,0
NO CHANGE ON ROTATION
SVD: 34, 8√5, 2√5, 0
EV(ATA ): 1156, 320, 20, 0
F803 has same EV’s
310
332
236
278
438
150
N.B. F803 has a different ATA matrix, but same EVs, SVD
378
212
206
360
370
368

32. Rotation of F109 group 1, pandiagonal8 11 14 15 10 5 4 6 3 16 9 12 13 2 7
Charpoly (Maple):
x(x-34)(x2-64)=0
eigenvalues: 34, ±8, 0
rank 3
SVD: 34, , , 0 12 6 15 1 13 3 10 8 2 16 5 11 7 9 4 14
34, 4√17, 2√ 17, 0
Rotated F109:
Charpoly:
x3(x-34)=0
eigenvalues : 34, 0, 0, 0

33. F175 & F790 group 3, regular 1 12 8 13 14 7 11 2 15 6 10 3 4 9 5 16
F175
EV’s : 34, ±8, 0
RF175: 34, ±8i, 0
SVD 34, 8√5, 2√5, 0
F790
EV: 34, 0, 0, 0
No change on rotation 5 4 16 9 11 14 2 7 10 15 3 6 8 1 13 12

34. F181 & F268 – nonsingular FULL RANK12 13 8 16 9 4 5 2 7 14 11 15 6 3 10
Group 11
F181: 34, -8,4±2i√2
RF181: 34, , ±5.3972i
SVD 34, , , 2 5 16 11 8 12 9 13 7 14 4 15 10 3 6
Group 7
F268: 34, , ,
RF268: 34, , ,
SVD 34, , ,
Do a more accurate SVD – ATA or Maple

35. Parameterization – Vector Spaces
Bergholt had 8 variables which reduce with further constraints.
For groups 1,2,3 we have found 4 dimensional spaces, and have used Maple to factorize their characteristic polynomials.
We have also found algebraic formulae for the eigenvalues of the Gramian matrix, ATA, which gives the squares of the SVD values, for groups 1 and 3.

36. Parameterization of Regular 4’s (48)
17-b
a+b+ c-17
b-a+d
34-d- c-b
17-c
17-a
a+c-d d
17-d
d-a- c+17 a c
b+c+ d-17
a-b- d+17
34-a -b-c b

37. Woodruff 1916 (n=8) x5(x-260)(x2-8736)=0 SVD 260, 129. 06, 72.032 34 63 37 60 6 27 48 49 56 41 23 10 20 13 19 14 52 45 55 42 24 9 62 35 29 4 26 7 57 40 25 8 58 39 61 36 30 3 51 46 11 22 44 53 47 50 16 17 38 59 5 28 2 31 33 64
Do a more accurate SVD – ATA or Maple

38. Regular, n=5 (Schindel, Trump)15 12 21 10 7 2 6 17 18 22 25 23 13 3 1 4 8 9 20 24 19 16 5 14 11
x2 (x-65)(x2-340)=0
SVD 65, , , [squared: 4225, , 200, ]
2007 Trump has studied all singular 5th order squares 2 11 21 23 8 16 14 7 6 22 25 17 13 9 1 4 20 19 12 10 18 3 5 15 24
x4(x-65)=0
SVD 65, , ,
Squared: 4225, 700, 434?, 165?
Do a more accurate SVD – ATA or Maple

39. Ultramagic, n=5 (Trump) Pandiagonal & regular After factor (x-65):
(x4-250x )
EV’s: 65, ±a, ±b
a =√(125-26√5)
b =√(125+26√5)
SVD 65, , , , 1 15 22 18 9 23 19 6 5 12 10 2 13 24 16 14 21 20 7 3 17 8 4 11 25
Do a more accurate SVD – ATA or Maple

40. Ultramagic, n=7 (Trump) 35 48 3 1 6 40 42 19 34 28 21 20 46 7 11 26 38 13 45 33 9 18 36 27 25 23 14 32 41 17 5 37 12 24 39 43 4 30 29 22 16 31 8 10 44 49 47 2 15
EV’s:
0,0,175,
±3, ±i√231
SVD 175, , , , , , 0
Do a more accurate SVD – ATA or Maple

41. Compound Squares Wayne Chan & Peter Loly, Mathematics Today 2002
Harm Derksen, Christian Eggermont, Arno van den Essen, Am. Math. Monthly (in press)
Matt Rempel, Wayne Chan, and Peter Loly
Adam Rogers’ Kronecker product

42. Compounded Lo-shu (1275 Yang Hui; Cammann)31 36 29 76 81 74 13 18 11 30 32 34 75 77 79 12 14 16 35 28 33 80 73 78 17 10 15 22 27 20 40 45 38 58 63 56 21 23 25 39 41 43 57 59 61 26 19 24 44 37 42 62 55 60 67 72 65 4 9 2 49 54 47 66 68 70 3 5 7 48 50 52 71 64 69 8 1 6 53 46 51
Do SVD

43. Second Compound Method (1275 Yang Hui; Cammann)31 76 13 36 81 18 29 74 11 22 40 58 27 45 63 20 38 56 67 4 49 72 9 54 65 2 47 30 75 12 32 77 14 34 79 16 21 39 57 23 41 59 25 43 61 66 3 48 68 5 50 70 7 52 35 80 17 28 73 10 33 78 15 26 44 62 19 37 55 24 42 60 71 8 53 64 1 46 69 6 51
Red: Lo-shu; Purple: incremented by 9; etc.
DO SVD

44. Kronecker Product For 2nd order A, any B Yucky typesetting!
You could use latex if you wanted to. :)

45. 2004 Adam Rogers (4th year Quantum Mechanics)
EN is Nth order square of 1’s
AM and BN are Mth and Nth order squares
Associative Compounding:
RA = EM  BN + Nk (AM  EN)
Distributive Compounding:
RD = BN  EM + Nk (EN  AM)
Given the EVs and SVDs of A and B, Rogers can find those for both compound methods
(k=2 for squares, 3 for cubes, etc.,)
Loly’s experiments have found ranks 5,6,7

46. “Franklin” binary 1 All 2x2’s sum to 2, as do all bent diagonals.
Half rows have sum 1, rows and columns sum 2. 1

47. Franklin Squares
At right – pandiagonal Franklin square PRSA
Arno van den Essen’s book
The 12th order question
[6May 2007]
Donald Morris n=12 “Franklin” 2007 1 32 38 59 5 28 34 63 46 51 9 24 42 55 13 20 27 6 64 33 31 2 60 37 56 41 19 14 52 45 23 10 11 22 48 49 15 18 44 53 40 57 3 30 36 61 7 26 17 16 54 43 21 12 50 47 62 35 25 8 58 39 29 4

48. Franklin, McClintock, Ollerenshaw & Brée
Bent diagonal squares, half row/column squares n = 8, 16
Complete squares 18..
Most-Perfect Pandiagonal squares
2006 exact count of Franklin squares for n = 8: 3*368,640 = 1,105,920
The problem of n = 12
22,295,347,200 complete squares (O&B 1998)
Eggermont – no pandiagonal F’s at n = 12
Donald Morris – 1/3 rows/cols for n = 12, 1/5 for n = 20, etc.

49. Inertia Tensor
Moment of Inertia of magic squares – Loly Math. Gaz. 2004
I = ∑mi (ri)2 => In = (1/12)n2(n4 -1)
Only need the semimagic constraints!
Inertia Tensor of Magic Cubes (Rogers and Loly, Am. J. Phys. 2004)
Folding magical squares to create magical cubes: 8*8 square => 4*4*4 cube
Kronecker products
Multiway arrays

50. Issues Constraint satisfaction problems (CSP’s)
Constraint Logic Programming (CLP), e.g., FormulaOne Compiler
Counting integer points in polyhedral cones (Maya Ahmed, Jesus de Loera, Matthias Beck, …)
Cryptography (O&B, Meine & Schuett)
Dither matrices (patents)

51. Conclusion Eigenvalues & SVD for small magic squares
SVD’s and “Music of the Squares”.
Compound squares (cubes, hypercubes)
Multimagic squares (Christian Boyer & Walter Trump)
Decoration – Claude Bragdon, architect
Applications – Chinese I Ching pattern!

52. Conjectures Normal magic squares have rank ≥3.
Normal non-magic pandiagonal squares have rank ≥2.
(Unless they are deflated to zero sum squares, in which case they are no longer normal.)

53. finis
Thank You

54. A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it sometimes happens that a kind of natural cunning and sagacity is of considerable value.
H.E. Dudeney