  We study the problem of arithmetic billiards from a new perspective. We first
raise a similar problem about reflecting lights inside grids, and give two
solutions to this problem. Next, we consider a similar problem in plane grids
and give its solution. Moreover, we extend this two problems to $p$-dimensional
space, where $p\geq2$. In this process, we introduce two mappings about finite
discrete sets, and get two finite abelian groups. In addition, we give the
definition of circular sequences and consider some combinatorial properties of
circular sequences.
