To say, "now on every clock tick we only take one step, not three," seems to imply that the optimized version takes 1/3 as many steps. The mammal still doesn't catch the reptile without doing two full laps while the reptile does one.<p>For an n-entry loop: The original version has the tortoise take n steps and the hare take 2n steps. The optimized version has the rabbit take 2n steps and the turtle take lg(n) steps.<p>For an n-entry line: The original version has the tortoise take n/2 steps and the hare take n steps. The optimized version has the rabbit take n steps and the turtle take lg(n) steps.<p>Or did I misread the algorithm?