Question 7
Let $k$ be a positive integer. Consider a game on an infinite one-dimensional board. Initially $n$ chips are placed in its cells, with multiple chips allowed in a cell. Each move is one of:
$(1^\circ)$ Choose two adjacent nonempty cells and move all chips from one to the other.
$(2^\circ)$ Choose a cell with at least two chips and move one chip $k$ cells left and one chip $k$ cells right.
a. If $n\leq k+1$, prove that only finitely many moves can occur.
b. For which values of $k$ can some $n$ and initial placement permit an infinite sequence of moves?
$(1^\circ)$ Choose two adjacent nonempty cells and move all chips from one to the other.
$(2^\circ)$ Choose a cell with at least two chips and move one chip $k$ cells left and one chip $k$ cells right.
a. If $n\leq k+1$, prove that only finitely many moves can occur.
b. For which values of $k$ can some $n$ and initial placement permit an infinite sequence of moves?