On a conjecture of Rademacher, Dickson, and Plotkin

AbstractGiven a closed n-dimensional cube in Euclidean n-space En, a box is (by definition) a closed rectangular parallelopiped contained in the cube and having edges parallel to the edges of the cube; a ray is a straight line of En which intersects the cube and is parallel to an edge of the cube; a configuration is a set of disjoint boxes which do not completely fill the cube although the set intersects every ray; a configuration is minimal if it consists of as few boxes as possible; and Kn is the number of boxes in a minimal configuration. Rademacher, Dickson, and Plotkin [1] conjectured that Kn=2n−1n and proved that K1=1, K2=4, K3=12, and K4≥30. We prove here that 2n−2(n+4)−2≤Kn≤2n−1n and state some further conjectures