## UIC Graduate Student Seminar

Friday September 28

3:00 pm

636 SEO

Speaker: Sam Hedayat

Title: Disproving The Euler Conjecture By Adding Latin Squares

Abstract: Surprisingly it is much easier to multiply mathematical objects than to add
them together.

I show you how to add Latin squares together for the purpose of obtaining
bigger Latin squares. In addition, I will show you if you carry this
addition over the finite fields then you can easily disprove the well-known
Euler conjecture (1723) which says there is no pair of orthogonal Latin
squares of order n if n is bigger than 6 and is in the form of n=2+4m. If
that is so easy, why then it took mathematicians over 230 years to disprove
the conjecture. The main reason being no one wanted to do the addition. I
will mention the joint work of R.C. Bose, S.S. Shirkahnde, and E.T. Parker
(often people refer to them as Euler spoilers), which disproved this
conjecture about 50 ago. But the conjecture could have been disproved much
much earlier if we knew how to add rather than multiply Latin squares.

Hundreds of papers have been written on the subject of Latin squares by
both mathematicians and statisticians because of their connection to finite
geometries, coding theory and experimental designs. I will give you some
key references and mention some research related problems.

For the mathematical aspects of Latin squares you may want to consult the
547 -page comprehensive book

Denes, J. and Keedwell, A.D. (1974). Latin Squares and Their Applications.
Academic Press, New York.