In the lastest issue of Emissary (the newsletter from MSRI) there's the following challenging puzzle: For every positive integer n, write it in binary, and allow the following possible set of transformations: You can insert "+" signs at arbitrary points within the binary expansion, and interpret that as a sum of binary numbers. For example 110101_2 --> 11_2+01_2+01_2 = 3 + 1 + 1 = 5 = 101_2. Show that there is an absolute constant C such that for any positive integer n there is a sequence of at most C such transformations that results in 1. Note: The minimal value of C is a real shocker This made me wonder about the obvious generalization to other bases, where now we ask for a sequence of such transformation that results in a single digit base b number. Victor