Entry #2 - The Folding Problem

The folding problem was quite tricky to wrap my head around. The idea of the problem is to find a way to predict the pattern of the directions of the creases on a piece of paper given the amount of times the paper was folded. I was able to get the first four patterns which gave me this result:

It occurred to me that on pattern 3, something was happening within it. Separating the pattern, I was able to see that pattern 2 and pattern 1 were in it. More importantly I noticed that pattern 2 was not only in pattern 3 twice, but the first time pattern 2 appeared, it had been flipped:

Removing pattern 1, I noticed that the other patterns followed the same format of flipping the previous pattern at the start, having a down arrow in the middle and then proceeding with the previous pattern as normal:

With this, I was able to come up with pattern 5:

With that in mind, I could very much just say that in order to predict the next pattern, start with the flipped version of the previous pattern, add a down arrow and then finish the normal version of the pattern. Looking at pattern 5 closer, I figured that there was another way to predict the patterns once I arranged it as so:

Pattern 3 is once again important to take into consideration since it outlines the patterns that follow it. The blue is the flipped version of the pink (or vice versa if it's more convenient to understand). It is then followed by a green arrow that separates the pattern. The direction in which the green arrow faces is dependent on the first pattern. 

To simplify this construct, let's create a sub-pattern which we will call a pair. One should write the blue pattern with a space and then write the pink pattern and consider it the pair. You repeat the pair 2^(x-3) times, x being the number that your pattern is. Then in order to fill in the spaces, you go to the first space on the left and then fill it in with the first arrow in the x-2 pattern. Keep doing this until both all the spaces are filled and you've finished using all the arrows in the x-2 pattern. Once that is done, the new pattern has been finished.

In summation: (U means up arrow, D means down arrow and S means space)

• Let x be the number of the pattern (Precondition: x >= 3)
• Let P(x) be the pattern itself (ex. P(2) = UDD)
• Let p be a pair in which UDDSUDD
• Repeat p 2^(x-3) times
• Replace the first S in P(x) with the first arrow in P(x-2) and follow along until all S have been replaced and all the arrows in P(x-2) have been used
• P(x) is completed.