### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Matthew Russell, Rutgers University, russell2 {at} math [dot] rutgers [dot] edu)
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Games For Arbitrarily Fat Rats

Speaker: Aviezri Fraenkel, Weizmann Institute of Science

Date: Thursday, November 5, 2015 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

In kindergarten we learned about the integers (Peano axioms); in grammar school about pairs of integers (rationals); and then in high school, about the reals (Dedekind cuts). Berlekamp, Conway, Guy discovered and promoted a method (Don Knuth: "Surreal Numbers") of creating all of those and much more namely games! in one masterful stroke.

Yet the rationals sometimes present obstinate difficulties often overlooked. Example. Let 1 < α₁ <, . . . , < αm be real numbers, dubbed moduli, m ≥ 3. An over 40 years old conjecture states that there exist reals γᵢ such that the system ( ⌊ nα₁ + γ ₁⌋ , . . . , ⌊nαm + γm⌋ ) constitutes a complementary system of m sequences of integers if and only if αᵢ = ⌊(2m - 1)/2m - i + γᵢ⌋ , i = 1, . . . , m. It is known that for integers and irrationals, 2 moduli have to be equal, but the problem is wide open for the rationals.

We have created, for every m ≥ 2, an invariant game whose P-positions (2nd player win positions) are the conjectured moduli, and gave game rules and an efficient strategy for the next winning move if not in a P-position. Motivation: (1) "Play" with the above conjecture. (2) Find efficient game rules for games defined only by their sets of P -positions. (Rats: rationals.)