Polygonal Rooms Not Illuminable from Every Point
George Tokarsky · American Mathematical Monthly · 1995
This problem has been attributed to Ernst Straus in the early 1950's, and has remained open for over forty years. It was first published by Victor Klee in 1969 [1]. It has since reappeared on various lists of unsolved problems, notably Klee again in 1979 [2] and in two recent books on unsolved problems, one by Klee and Wagon in 1991 [3] and one by Croft, Falconer and Guy, also in 1991 [4]. In this article, we will settle the above problem in the negative. We will as well give elementary techniques for constructing rooms, both in the plane and in three-space, which are not illuminable from every point. In particular, we will show that if the two people are located in a two-dimensional planar room as shown in Figure 2, then they cannot see each other.