Tiling with Incomparable Cuboids
Charles H. Jepsen · Mathematics Magazine · 1986
In this note we answer three questions raised by Richard K. Guy in his column Unsolved Problems in The American Mathematical Monthly of December 1984 [1]. Guy discusses tiling a three-dimensional cuboid using (at least two) incomparable cuboids with integer sides. Two cuboids are incomparable if neither will fit inside the other with sides parallel. (To compare two cuboids, write their dimensions in increasing order: a1 K a2 ?. a3, b1 b1.) Bill Sands found a tiling of a 3 x 4 x 15 cuboid (volume 180) using the six pieces 1 x 1 x 15, 1 X 2 X 11, 1 X 3 X 10, 2 X 3 X 6, 2 x 4 x 4, 3 x 3 x 5. (See FIGURE 1).