2012/06/20 by Larsson, Urban
#11B75 #91A46 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1206.4485
We study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens (x,y). Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the game terminates when a player in turn is unable to move. We follow the normal play convention, which means that a player who cannot move loses. In the game of Wythoff Nim, a player is allowed to remove either any number of tokens from precisely one of the piles or the same number of tokens from both. Let ϕ= (1+√(5))/(2) and for all nonnegative integers n, An=\lfloorϕn \rfloor and Bn=An+n. The P-positions of Wythoff Nim are all pairs of piles with An and Bn tokens respectively. We study a generalization of this game called (1,2)\G where, in addition to the rules of Wythoff Nim, a player has the choice to remove a positive number of tokens from one of the piles and twice that number from the other pile. We show that there is an infinite sector α≤ y/x ≤ α+ε, for given real numbers α>1 and ε> 0, for which each (x,y) is an N-position, but that there are infinitely many P-positions for both 1≤ y/x 0, is a pair of so-called complementary sequences on the natural numbers which satisfy (ai) is increasing and for all i, ai