1991 CMO 第 6 题

A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows:

i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread;

ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors.

Show that we can assign to each node on the ball a complex number (not equal to $1$ ), such that the product of the numbers assigned to the vertices of any polygonal face is equal to $1$ .

足球上覆盖着一些多边形的皮革片，这些皮革片由三种不同颜色的线缝制而成。其特点如下：

i)用某种颜色的线将一块多边形皮革的任意一条边与另一块多边形皮革的等长边缝合起来；

ii) 球上的每个节点恰好是三个多边形的顶点，并且该节点处的三个螺纹接头具有不同的颜色。

证明我们可以为球上的每个节点分配一个复数(不等于 $1$ )，使得分配给任何多边形面的顶点的数字的乘积等于 $1$ 。