Penny Piles

On the March 13th lecture, we spent the day working with partners trying to figure out whether we can arrange two piles of pennies in order to get every number in the range of [0, 64]. The way me and my partners did this was by reducing the range to [0, 32]. We knew that if we could create piles of pennies that had every number in this range, than it would be true of the range [0, 64] as well. So we proceeded by using a brute force method and calculated that every number in the range [0, 32] could be created by using a combination of dividing the left and right piles. The simplest way this could be done would be to work backwards. For example, the combination (13, 19) could be created from this progression:

(13, 19), (26, 6), (20, 12), (8, 24), (16, 16), (0, 32). 

First we would divide the pile a few times, then work backwards, and attempt to fill in the gap between the two parts. This does seem like a good approach, but perhaps not the best one. At first I did not think this would be possible for all numbers in the range, [0, 32], but it turns out I was wrong. I also do not really see a different way of doing this, but I do remember different methods being discussed at the end of lecture, I just don't really remember what they were. A tree diagram was also mentioned on the handout, but I just figured that would be a different way to organize the results while still using the brute method approach. Overall, I think this was an interesting problem, and one that I did not expect to be true as well.