Polygonal Rooms Not Illuminable from Every Point

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.