Knuth has put his Boolean Synthesis work online in a tarball at www-cs-faculty.stanford.edu/~knuth/programs/boolchains.tgz He managed to crack one of the 7 hard functions, leaving 6 to go. My own effort at a synthesis search program isn't doing very well so far -- it runs all night on small test cases, and frequently doesn't find any synthesis at all, etc. etc. --------- Digit silliness: Re the question of whether all sans-0-digit numbers appear in the closure of {2}, a related question occurred to me: Do all such sans-0-digit strings appear somewhere in the square of a sans-0-digit number? Statistically, it seems unlikely that most such strings appear in the square of a *shorter* s0d number. This would mean that, if all s0d numbers occur in the closure of {2}, most of them are reached through a path that involves shrinking down from a larger number. Second question: Do all the s0d numbers generate the same set? Or, Do all s0d numbers generate 2? Rich