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.