论文标题
所有蚱hopper都在哪里？
Where have all the grasshoppers gone?
论文作者
论文摘要
令$ p $为飞机上的$ n $ element点。考虑$ n $（点状的）蚱hoppers位于$ p $的不同点。在“法律”举动中，他们中的任何一个都可以跳过另一个，并以完全相同的距离降落在另一侧。经过有限的法律动作，蚱hoppers可以在类似但大于$ p $的点套装中最终出现吗？我们提出了一种线性代数方法来回答这个问题。特别是，如果$ p $是常规$ n $ -gon和$ n \ neq 3、4、6 $的顶点集，我们通过证明答案是肯定的，解决了布鲁克的问题。还考虑了一些概括。
Let $P$ be an $N$-element point set in the plane. Consider $N$ (pointlike) grasshoppers sitting at different points of $P$. In a "legal" move, any one of them can jump over another, and land on its other side at exactly the same distance. After a finite number of legal moves, can the grasshoppers end up at a point set, similar to, but larger than $P$? We present a linear algebraic approach to answer this question. In particular, we solve a problem of Brunck by showing that the answer is yes if $P$ is the vertex set of a regular $N$-gon and $N\neq 3, 4, 6$. Some generalizations are also considered.
