Lower bounds for the 'cycle detection problem' were recently investigated by Fich (1981, 1983). She showed that Floyd's algorithm was optimal among those algorithms which have M = 2 memory locations and which make a finite number of 'jumps'. A lower bound for the case where M > 2 was also presented, but the question of whether having more than two memory locations could actually yield a better algorithm was left open. In this report, we show that it cannot. A lower bound was also presented by Fich (1981, 1983) for algorithms which have two memory locations and which make a finite number of 'back advances'. We show here that the same lower bound holds even if the restriction on back advances is dropped. © 1985.