Kerry Mitchell: My initial calculations show that the orbit of 0 under "add 1 and reverse bits" is periodic: - if the number of bits (m) is odd, then the period is 2^m, - if the number of bits is even, then the period is 2^(m/2+1)-1. Verified from m=1 to 30 bits. --WDS: the m=even case seems provable: X is always a palindrome after an even #steps starting from X=0. Until finally we reach 1111..111 which takes (2^(m/2)-1)*2 steps, then next step yields 0. QED. the m=odd case is trickier.