Hong-Yeop Song hysong@yonsei ac kr hysong@yonsei.ac.kr - PowerPoint PPT Presentation

Ch i' Choi's orthogonal Latin Squares th l L ti S is at least 67 years earlier than y Euler's 2008 Global KMS International Conference 2008. 10. 23.~10. 25. JEJU ICC, KOREA Hong-Yeop Song hysong@yonsei ac kr hysong@yonsei.ac.kr http://coding.yonsei.ac.kr School of Electrical and Electronic Engineering g g Yonsei University, Seoul, KOREA

Content Content ☞ Prelude l d ☞ Magic Squares M i S ☞ Latin Squares and Orthogonality L i S d O h li ☞ Orthogonal Latin Squares and Magic Squares O h l L i S d M i S ☞ Summary S

Prelude l d Seok-Jeong Choi Seok-Jeong Choi Leonhard Paul Euler Sang Geun Hahn Hong-Yeop Song Hong Yeop Song

최석정 ( 崔錫鼎 , 1646년~1715년)은 조선 후기 의 문신이다. 여러 요직을 거쳤고 1701년 영 Leonhard Paul Euler (1707 – 1783) 의정이 되었다 그는 수학서 《구수략》 을 저술 의정이 되었다. 그는 수학서 《구수략》 을 저술 was a pioneering Swiss was a pioneering Swiss 했는데 이는 그가 1710년 영의정을 그만둔 이 mathematician and physicist who 후 작성했을 것으로 추측된다. spent most of his life in Russia and C o , Seo Choi, Seok-Jeong (1646-1715) has served Yi Jeo g ( 6 6 5) as se ed Germany. Euler is considered to be Germany Euler is considered to be Dynasty in various positions, including the the preeminent mathematician of prime minister in 1701. He wrote Koo-Soo- the 18th century and one of the Ryak which was believed to be written y greatest of all time . greatest of all time after he retired in 1710. Pair of orthogonal orthogonal Latin squares 1782 1715 67 years?? 67 years??

From Mathematics Genealogy Project North Dakota State University, http://genealogy.math.ndsu.nodak.edu � Hong-Yeop Song , Ph.D. University of Southern California 1991, USA, Advisor: Solomon W Golomb Advisor: Solomon W. Golomb � Solomon Wolf Golomb , Ph.D. Harvard University 1957, USA, Advisor: David Widder � David Vernon Widder , Ph.D. Harvard University 1924, USA, Advisor: George Birkhoff � George David Birkhoff , Ph.D. University of Chicago 1907, USA, Advisor: E. H. Moore � E. H. (Eliakim Hastings) Moore , Ph.D. Yale University 1885, USA, Advisor: H. A. Newton � H. A. (Hubert Anson) Newton , B.S. Yale University 1850, USA, Advisor: Michel Chasles � Michel Chasles , Ph.D. Ecole Polytechnique 1814, France, Advisor: Simeon Poisson � Simeon Denis Poisson France � Simeon Denis Poisson , France, Advisor: Joseph Lagrange Advisor: Joseph Lagrange � Joseph Louis Lagrange , Advisor: Leonhard Euler

PART 1: Pre-history ~1991 ~1991 H YSong (Grad student at USC) has met Dr Denes (co-author of “Latin H.Y.Song (Grad student at USC) has met Dr. Denes (co author of Latin Squares and Their Applications” 1974) who was a good friend of his advisor Dr. Golomb. ~1993 ~1993 S.G.Hahn (Prof. at KAIST Math dept.) has seen the original printing of S G Hahn (Prof at KAIST Math dept ) has seen the original printing of Koo-Soo-Ryak by Choi Seok-Jeong, and published a (Korean) paper on KSESM journal about the original Pair of Orthogonal Latin Squares of order 9 He also had some correspondents with Dr Denes about this order 9 . He also had some correspondents with Dr. Denes about this . 1994 H.Y.Song (Senior Engineer at Qualcomm) published a contribution to CRC Handbook of Combinatorial Designs, co-authored with one of co- editors, J. Dinitz (Prof. at U. Vermont).

PART 2: how it begins 1997 1997 H. Y. Song (Prof. at Yonsei Univ.) H Y S (P f t Y i U i ) ● met Dr. Denes at Ulm, Germany, attending IEEE ISIT, and he was asked to find some Korean journal paper; ● was SO SURPRISED to see the paper by S G Hahn sent the paper ● was SO SURPRISED to see the paper by S. G. Hahn, sent the paper and its English translation to Dr. Denes; ● made a (first) call to Prof. Hahn asking a copy of the original printing. 1998 Prof. Hahn published a newsletter contribution to KSESM about this .

PART 3: how it ends Dr. Denes has passed away. He was not able to publish the 2 nd edition 2001(?) of his book, Latin Squares and Their Applications . We guess he was planning to say something about Choi’s POLS there was planning to say something about Choi s POLS there. 2005 H. Y. Song ● was asked to revised his contribution to CRC Handbook of ● was asked to revised his contribution to CRC Handbook of Combinatorial Designs for 2 nd edition; ● has asked to see the galley version of a newly added section on “History of Combinatorial mathematics” and decided it is the time; History of Combinatorial mathematics , and decided it is the time; ● sent all he had so far to Dr. Dinitz.

The page of the book discussing Choi’s POLS of order 9

The page of the book showing Choi’s birth place and year.

Pair of orthogonal Latin squares of order 9 Koo-Soo-Ryak, 1715(?), written by Seok-Jeong Choi (최석정). Koo-Soo-Ryak, 2006, translated by Hae-Nam Chung and Min Heo (정해남∙허민, 교우사)

Choi has constructed the following POLS of order 9 to form a magic square. Pictures from Monthly Mag. of Dong-Ah Sciences, Aug. 2008, Article by S.K. Kang

Handwritten copy (early 1700?) – Library of Yonsei University C.H. Lee (Prof. Math, Yonsei University) has donated this in the 1950s.

Handwritten copy (early 1700?) – Library of Yonsei University

Magic Squares Magic Squares

A magic square of order n with magic constant d is an n ⅹ n array of n 2 g q g y integers such that 1. Every row-sum is d 2. Every column-sum is d y 3. LR diagonal-sum is d 4. RL diagonal-sum is d. A semi-magic square satisfies only the conditions 1 and 2 above. A magic square of order n is called normal if the entries are 1,2,…,n 2 . There are numerous ways to generalize the magic squares: magic circles, magic cubes, magic graphs, magic series, magic tesseracts, magic hexagons magic diamonds magic domino tilings magic stars magic hexagons, magic diamonds, magic domino tilings, magic stars, bimagic squares, trimagic squares, etc. See http://mathworld.wolfram.com/topics/MagicFigures.html See ttp:// at o d. o a .co /top cs/ ag c gu es. t Or http://www.multimagie.com/indexengl.htm

Picture from Mathematics Concert (수학 콘서트, 박경미, 동아시아 2006) Lo Shu [ 洛書 ] (낙서) Some three thousands years ago in China, when King y g , g tried to appease the river god, a turtle with the following pattern of numbers on its back had appeared…..

For an odd integer n≥1, there is an “easy” construction for a (normal) magic square of order n. Example of order 5 Example of order 5

RU corner of Melencolia I Melencolia I Engraving (1514) Albrecht Dürer (1471∼1528) Albrecht Dürer (1471 1528) It has the additional property that It h th dditi l t th t � the sums in any of the four quadrants ( composite magic square), � as well as the sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). It is thus a gnomon magic square. In addition, any pair of numbers symmetrically placed about the cen y p y y p ter of the square sums to 17, a property making the square even m ore magical.

Picture from Mathematics Concert (수학 콘서트, 박경미, 동아시아 2006) Magic square of order 4 (non-normal) by Srinivasa Ramanujan (1887 – 1920, India) 22 22 12 12 18 87 18 87 21 21 84 84 32 32 2 2 92 92 16 16 7 7 24 24 4 27 82 26 Magic constant = 139 Magic constant = 139

Latin Squares Latin Squares

A latin square of order n is an n ⅹ n array of n different symbols such that each row and each column is a permutation of these n symbols. 1 1 2 1 2 3 1 2 3 1 2 3 4 1 2 3 4 2 1 3 1 2 2 3 1 2 1 4 3 3 4 1 2 2 3 1 3 1 2 3 4 1 2 4 3 2 1 4 3 2 1 2 1 4 3 A transversal in a latin square of order n is a set of n cells (or positions), one from each row and column, containing each of n symbols exactly once. Two latin squares A=(a ij ) and B=(b ij ) of order n are called a pair of orthogonal latin squares (POLS or a graeco-latin square ) of order n orthogonal latin squares (POLS, or a graeco latin square ) of order n if the n 2 ordered pairs (a ij ,b ij ) are all distinct for 1≤i,j≤n. In this case, we say that A is orthogonal to B or vice versa. α β γ a b c αa βb γc γ γ α α β β b b c a c a γb γb αc βa αc βa c a b β c γa αb β γ α Theorem: A Latin square of order n has an orthogonal mate if and only if it can be decomposed into n disjoint transversals.