2178 and All

For integers g ≥ 2, k ≥ 2, call a number N a (g, k)-reverse multiple if the reversal of N in base g is equal to k times N . The numbers 1089 and 2178 are the two smallest (10, k)reverse multiples, their reversals being 9801 = 9 · 1089 and 8712 = 4 · 2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g, k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfermatrix method to enumerate the (g, k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood.