1. “Origins” Board games on a square grid – Sudoku, chess, and magic squares – from refereed papers to personal web pages or vice versa.
Peter Loly, Senior Scholar, Physics and Astronomy, University of Manitoba.
UNIVERSITY OF ICELAND –UNIVERSITY OF MANITOBA Partnership Conference to be held at the University of Iceland, Reykjavík, August 23–24

2. Vacation Scholarship 1962 – Queen Mary College, London

3. The Manitoba Magicians [with apologies to the International Brotherhood of Magicians (IBM), founded in Winnipeg in 1922]
John Hendricks, mathematician, meteorologist (retired to Victoria, British Columbia) (September 4, July 7, 2007)
Frank Hruska, Chemistry, University of Manitoba, Senior Scholar,
Vaclav Linek and John Cormie (antimagic squares) University of Winnipeg
Peter Loly, retired December 2006, Senior Scholar, U Manitoba
Ian Cameron, Director, Planetarium and Observatory , U Manitoba
Marcus Steeds, Frantic Films
Wayne Chan, Centre for Earth Observation Science (CEOS)
Adam Rogers, Ph.D. 2012, genetic algorithms for astrophysics
Daniel Schindel, Michigan State, East Lansing, nuclear theory, now at University of Western Ontario.
Matthew Rempel, second degree in engineering
Honorary members: Walter Trump, Nuremberg, and
George Styan, Prof. Emeritus, Mathematics and Statistics, McGill.
Red and Blue: undergraduates at U Manitoba; Red are published with PDL.

4. The earliest known appearance of this square was found in the ruins of Pompeii which was buried in the ash of Mt. Vesuvius in 79 AD. Also at Corinium (modern Cirencester in England).

5. Chess Moves
Castles (rooks) can only move along Rows (chess: ranks) or Columns (chess: files).
Bishops only along Diagonal paths.
The Queen can roam along Rows, Columns and Diagonals.
Knights move one row and two columns, or vice versa.
Some magic squares exhibit a continuing sequence of knight’s moves which cover the entire board.

6. Thanks to Sid Kroeker, archaeologist, Winnipeg Philatelic Society, pocket stamp dealer: chess checkerboard

7. Squares 4 corners 4 edges 2 diagonals (main and dexter)
8 directions of the wind,
N,S,E,W and NE,NW,SE,SW
Buddhism’s eightfold way,
AND OF COURSE THE ORIGIN IS AT THE CENTRE!!!

9. Teaching the physics of crystals: 16. 452, 16
Teaching the physics of crystals: , The ORIGIN and the Cartesian axes!

10. Myers-Briggs Personality Square (MBTI)
Katharine E. Cook, b. 1875;
marries Lyman J. Briggs 1896
Daughter: Isabel M. Briggs b. 1897;
marries C. G. Myers 1918
Carl Jung 1920 Psychological Types; trans. 1923
MBTI: Katherine and Isabel 1943
Myers, Isabel Briggs with Peter B. Myers, (Original edition 1980)

14. P. D. Loly. "Quarto Plus!" Bulletin of the Association for Psychological Type, 20(2):45, 1997.
Quarto! has a 4x4 board and 16 pieces. Each piece has four dichotomous attributes: color, height, shape and consistency. So each piece is either black or white, tall or short, square or round, and hollow or solid. The object is to place the fourth piece in a row where all four pieces have at least one attribute in common. The twist is your opponent gets to choose the piece you place on the board every turn.

16. Personality Types (MBTI )
MBTI Training mid-90’s – qualified with supplementary essay
PDL audited 4H/graduate course in 1997 with Marianne Johnson, Psychology
Essay on “3”.
4-bit coding of combinations of 4 dichotomous dimensions! “4D”
P. D. Loly. "A purely pandiagonal 4*4 square and the Myers-Briggs Type Table." Journal of Recreational Mathematics, 31(1), 29-31, 2000/2001 (accepted 1998).
Used Leibniz’s (1646–1716) binary arithmetic (c. 1679, pub. 1703)
[Thomas Harriot (c ) – binary math a century before Leibniz.
Chinese yin-yang binary system much older.
And Egyptian peasant math 18C BCE.]

17. Loly’s J.Rec.Math
John Vukelic et al in , c :
> Karnaugh maps
NOT a magic square!
Natural magic squares of order n have same sum for all Rows, Columns and both Diagonals (“RCD”). 8 6 2 4 7 5 1 3 15 13 9 11 16 14 10 12
Logic square initially found from a matrix of personality types – Myers-Briggs (Carl Jung).

18. I Ching
hexagrams
Yin, 0,
broken line;
Yang, 1,
solid line.
6-bits, 2^6=64
Fold to a cube!
Extension:
8-bits -> 256

19. pp. 2-13: SCHOLAR

20. Latin Squares Each row and column has just one of n symbols.
E.g., 1..n, or 0,..(n-1).
CONTRAST: the usual context for magic squares requires distinct elements, with natural magic squares having elements 1..n2.
Or, 0,..(n2-1).

21. 1972 for International Congresses: Geology, Geographic, Cartography and Photogrammetry (mini-Sudoku)
WPS. C.2006/7

22. Sudoku
On a 9-by-9 board with 81 cells it suffices to begin with just 17 initially filled in order to obtain the full solution.
Each Row and Column must contain one of the integers 1..9,
Which would make it a Latin square,
BUT...
Each 3-by-3 subsquare (tile) must hold them all.
1972 – the first modern Sudoku puzzle.
Boyer finds an earlier French origin.
Euler 18thC; Korea?

23. Potential project

24. Associative or regular MS: antipodal constraint (local).

25. The First Magic Square
A 3-by-3 arrangement of 1..9 such that all Rows, Columns and BOTH Diagonals add (sum) to the same magic constant:
Sum all 9 cells: =45.
Each straight line of 3 cells must then add to 45/3=15.
And the average is 15/3=5 (the centre cell).
The Loshu (Luo-shu, ...) dates to c. 600 BCE.
The motif for Feng Shui – 8 directions (eight-fold Way)
Marie-Louise von Franz 1974 “Number and Time”

26. Each row (3), column (3) and diagonal (2) [8 sets] adds up to 15.
All the numbers add up to 45.
The average of the 9 numbers is 5 - the centre - balance! [Centre of mass]
The number of constraints equals the number of different triples from 1..9.

27. Traced from a painting by Christy Belcourt,
Traced from a painting by Christy Belcourt, Manitoulin Island, Ontario
Traced from a painting by Christy Belcourt,
Manitoulin Island, Ontario

28. Larger Magic Squares
There are 880 distinct natural magic squares using 1,2,3,...16(=4*4) on a 4-by-4 board.
Bernard Frenicle de Bessy (1693) – he died in 1675.
In 1972 Schroeppel used a PDP computer to show that there are 250,305,224 distinct magic squares on a 5-by-5 board.
Estimates for on a 6-by-6 board ...

30. Peter’s 70th cake! (linesum 65)

31. Magic Square Counts: Walter Trump (c. 2002)
semi-magic B:
natural 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?).

32. Populations c. 10^82 Ollerenshaw & Bree formula for n=4p. E.g. n=72
A few known exactly
n=3,4,5
Franklin squares at n=8: 3*368,640 by Schindel, Rempel, Loly in 2006
Ollerenshaw & Bree formula for n=4p.
E.g. n=72
About 5*10^107
Atoms in the Universe
c. 10^82
by John Carl Villanueva on July 30, 2009
Read more:
Walter Trump 2007: estimate for 10*10 magic squares: (12) ·10110

33. Amateurs
Astronomy: Canadian David Levy and the Shoemaker-Levy comet of 1993/4 which hit Jupiter.
SN1997A - Ian Shelton, Winnipeg, Manitoba, Toronto, Chile.
Sudoku and chess are both examples of recreational mathematics which link with the broader issue of magical squares.

34. En.wikipedia.org: mid-Leonardo-Flywheel.ogg.jpg

35. Laws of Motion Newton’s 1st law
Rectilinear motion in absence of force(s)
Newton’s 2nd law
“F=ma”
Newton’s 3rd law
Action and reaction at contact
AND their rotational counterparts
Mass->moment of inertia
Linear momentum-> angular momentum

36. Moment of Inertia & Inertia Tensor
The Mathematical Gazette 88, , 2004
Theorem for magic squares: I(n)=n^2(n^4-1)/12
********ORIGINALITY**********
Adam Rogers, summer student 2003, extended to cubes:
Another theorem: Am. J. Phys. 72, 786-9, 2004
David Politzer
N.L. 2004
Both theorems indexed in Sloane’s OEIS.

37. “Franklin151” 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

38. Franklin squares on 8-by-8 board
Ben’s 300th anniversary in 2006
Handful of “bent-diagonal” semi-magic squares c. 1769
Dan Brown, “The Lost Symbol”, 2009, p.337 “Eight Franklin Square”, and p.389 [semimagic, R=463,223,040]
Counted by Schindel, Rempel and Loly in 2004, published in Proc. Roy. Soc. A in 2006 – media coverage in Canada, U.S. and Holland.
********ORIGINALITY**********
Total: 1,105,920=3*368,640 exactly. <<90 clans>>
368,640 pandiagonal magic squares. <<10 clans>>
2*368,640 semi-magic
USE: target to test efficiency of computer programs for difficult problems
Turning point in career – now invited to give keynote talks at international conferences
R: invention?

40. Knight’s Tours Islamic: 840, 920, Al-Adli, c. 1200 CE
Faulhaber and Descartes met in
Euler ( ) found a complete Knight’s tour on a semi-magic square.
Feisthamel (1884) has an almost complete tour on a magic square except for the gap between 32 and 33:
See Jan Gullberg, “Mathematics From the Birth of Numbers“, 1997.
Knight’s paths – a different “game” where a knight move progresses at a fixed angle across a tiled chess board – George Styan, McGill.

41. Compounded Lo-shu Abū'l-Wafā al-Būzjanī [940-997/8 A. D
Compounded Lo-shu Abū'l-Wafā al-Būzjanī [ /8 A.D.; see Sesiano 1996 and 2004] 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

42. 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

43. MAP: IWMS 2007 Windsor, ON, & Lin. Alg. Appl. 430 (2009) 2659-2680
Realization that SVD analysis resolved problems with eigenvalue analysis.
SVD=Singular Value Decomposition
The SVs are non-negative and decreasing.
The larger values are most significant, while the smaller values may be insignificant or zero.
For small integer matrices the SVs are often pretty.
Google page rank algorithm noted at IWMS2007

44. Shannon Information\Entropy
Claude Shannon, 1947,
Bell Systems Technical Journal
Messages: transmitter->channel->receiver
John Von Neumann
Leon Brillouin
CONTENT ANALYSIS?
Tom Carney, UM ‘70’s

45. Shannon Entropy and Magical Squares
Newton, P.K. & De Salvo, S.A. (2010) [“NDS”]
“The Shannon entropy of Sudoku matrices”, Proc. Royal Soc. A 466: [Online Feb ]
Immediately clear to us that we could extend NDS using our studies of the singular values of magic squares in Loly, Cameron, Trump and Schindel in LAA 2009.
********ORIGINALITY**********

46. Indus Script – Rajesh Rao – TED talk – Rao et al Science 324 (2009) 1165; Pauline Ziman et al

47. Shannon entropies for MATLAB’s magic(n): n odd, even, doubly even

48. Albert Girard 1629 Fundamental Theorem of Algebra.
Polynomial of degree n has n roots!
Some may be complex, others be zero.
Formulae for sum of powers of roots – symmetric functions.
Loly (August 2010): if coefficients are integer, as in magical squares, then these sums of powers are also integer, AND
generalizing Girard’s “sums” leads to an INTEGER INDEX for ALL integer squares, not just magical ones!!!
***ORIGINALITY***CAREER HIGH POINT

49. The Library of Magical Squares
Analyze historical squares
Place newly constructed squares by engaging amateurs and recreational mathematicians
Library of Celsus at Ephesus.

50. Libraries, Librarians and Archivists
UM: Bill Poluha, Science; Shelley Sweeney Archives; Barbara Bennell (rel. John Hendricks), Document Delivery; Norma Ghodavari, Engineering; ...
(Picture: Library of Celsus, Epheseus)
U. Louisiana at Shreveport archives (Frierson)
Strens-Guy archive of Recreational Mathematics, U. Calgary, Appolonia Steele.

51. Mathematics education in Iceland
Kristin Bjarnadottir, UI Education
Paper: “FUNDAMENTAL REASONS FOR MATHEMATICS EDUCATION IN ICELAND”
PDL: speculates that there may be books and archived journals in Iceland containing chess magical squares.
E.g. Edward Falkener, “Games Ancient & Oriental & How to Play Them ”, c
British Chess Magazine, present

52. Dame Kathleen Ollerenshaw, b. 2 October 1912
Dame Kathleen Ollerenshaw, b. 2 October Monograph with David Bree, 1998.
Dame Kathleen Ollerenshaw, b. 2 October 1912.

53. Water Retention by Craig Knecht
An order-9 simple magic square with 1 lake, 2 ‘ponds’ and 3 ‘islands’. 780 units are retained. [4]
[4] These 3 squares were constructed by Walter Trump. He has a web site at

54. The Icing on the Cake

55. Google Ngram results from 500 million digitized books