User blog comment:LittlePeng9/ITTM galore/@comment-5529393-20140318223404

Using LittlePeng9's "master flag" idea, here's a five state, three symbol program for w^w:

0 _ _ r 1 1 _ 1 l 2 1 1 2 r 4 1 2 1 r 1 2 _ 1 l 3 3 _ _ l 3 3 1 _ l 3 4 _ _ l 4 4 1 1 l 4 4 2 1 l 4 L 0 0 r 1 L 1 1 r halt

basically, each cell (after the first) goes to 1 the first time we visit it, then flashes from 1 to 2. The second cell flashes every w steps, so every w^2 steps the limit stage will have the second cell be 2, and we will change the third cell from 0 to 1, then flash it from 1 to 2. Then, every w^3 stages the third cell will be 2, so we will move the fourth cell, and so on. Each time we go from 0 to 1, we go back to the first cell, and flash it from 0 to 1 and back again. Finally, after w^w steps, the first cell will have value 1, and the ITTM halts.