Just was wondering if this group might have anything to add about this 1972 puzzle? Can you cite other elegant questions, with such startling special-case answers? ======== PROBLEM 45 (Gosper): Take a unit step at some heading (angle). Double the angle, step again. Redouble, step, etc. For what initial heading angles is your locus bounded? PARTIAL ANSWER (Schroeppel, Gosper): When the initial angle is a rational multiple of pi, it seems that your locus is bounded (in fact, eventually periodic) iff the denominator contains as a factor the square of an odd prime other than 1093 and 3511, which must occur at least cubed. (This is related to the fact that 1093 and 3511 are the only known primes satisfying P 2 2 = 2 mod P ). But a denominator of 171 = 9 * 19 never loops, probably because 9 divides phi(19). Similarly for 9009 and 2525. Can someone construct an irrational multiple of pi with a bounded locus? Do such angles form a set of measure zero in the reals, even though the "measure" in the rationals is about .155? About .155 = the fraction of rationals with denominators containing odd primes squared = 1 - product over odd primes of 1 - 1/P(P + 1). This product = .84533064 +- a smidgen, and is not, alas, sqrt(pi/2) ARCERF(1/4) = .84534756. This errs by 16 times the correction factor one expects for 1093 and 3511, and is not even salvaged by the hypothesis that all primes > a million satisfy the congruence. It might, however, be salvaged by quantities like 171.