Because this activity was so popular the first time around, I wrote more codes. The first column can be done on pegboard. It is better on graph or dotted paper. I did make a couple of mistake in the code and they are now corrected.
Supplies: one or two sheets of graph paper/dotted paper. You can print some off from https://incompetech.com/graphpaper/ EDITED TO ADD: for my 5-year old, I found that using the pegboard as a grid and drawing the path with markers on it works well.
If you scroll down a little on the website linked above, the pattern paper looks really cool. I discovered from the Fractal activity that the kids love having the triangle patterns to color however they like.
A riddle (sadly relevant)
A lily pad sits on a pond. It doubles in size every day. It takes 8 days for it to cover the pond.
Question 1: How many days does it take for it to cover half the pond?
Question 2: If you start with 4 lily pads instead, how many days does it take to cover the pond?