Complementary Iterated Floor Words and the Flora Game

Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ2. We use them to formulate an algebraic polynomial-time winning strategy for a new four-pile take-away game Flora, which is motivated by partitioning the set of games into subsets CompGames and PrimGames. We further formulate recursive, arithmetic, and word-mapping winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. We further show how to generate the floor words induced by the iterations using word-mappings and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences. Key words. floor function, integer part function, combinatorics of words, combinatorial game theory, Fibonacci numeration system AMS subject classifications. 11B75, 11B39, 91A46, 05A05 DOI. 10.1137/090758994

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Aviezri S. Fraenkel
• Keywords: Fibonacci Numeration, Polynomial-time Winning Strategy, SIAMDM 2010, floor Function