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.

翻译：我们考虑斯宾塞于1977年提出的平衡游戏的一个特殊几何情形。设平面上有n条直线的任意构型$\mathcal{L}$，并假设构型的每个单元内均放置一个盒子。爱丽丝初始时在每个盒子中放置石子。在后续每一步中，鲍勃选择一条直线，爱丽丝必须选定该直线的一侧，移除该侧每个盒子中的一颗石子，并在另一侧每个盒子中添加一颗石子。若任一盒子变空，则鲍勃获胜。我们确定了多项式时间内可计算的最小石子数$f(\mathcal L)$——在此数量下爱丽丝能阻止鲍勃获胜，并证明对于任意处于一般位置的n条直线构型$\mathcal{L}$，均有$f(\mathcal L)=Θ(n^3)$。