Geometric Give and Take
This work provides a theoretical bound for a specific geometric balancing game, which is an incremental contribution to combinatorial game theory.
This paper analyzes a geometric balancing game involving pebbles in boxes within line arrangements. It determines the minimum number of pebbles Alice needs to prevent Bob from winning, finding this number is computable in polynomial time and is Θ(n^3) for n lines in general position.
We consider a special, geometric case of a balancing game introduced by Spencer in 1977. Consider any arrangement $\mathcal{L}$ of $n$ lines in the plane, and assume that each cell of the arrangement contains a box. Alice initially places pebbles in each box. In each subsequent step, Bob picks a line, and Alice must choose a side of that line, remove one pebble from each box on that side, and add one pebble to each box on the other side. Bob wins if any box ever becomes empty. We determine the minimum number $f(\mathcal L)$ of pebbles, computable in polynomial time, for which Alice can prevent Bob from ever winning, and we show that $f(\mathcal L)=Θ(n^3)$ for any arrangement $\mathcal{L}$ of $n$ lines in general position.