Notes: 61.25 Prescribed numbers of tiles and tilings

Extract

In their recent article [1] B. Grünbaum and G. C. Shephard asked whether, for every pair of natural numbers k, r, there exist sets of k tiles that admit precisely r distinct tilings of the plane. (Two tilings are said to be distinct if they cannot be brought into coincidence by a rigid motion, and it is to be understood that in the tilings each distinct tile must occur at least once.) They gave examples for k = 1, r = 1, 2 and for k = 2, r = 1, and referred in a footnote to a later discovery of a solution for k = 1, r = 3.