Paper Folding problem

At the start of the semester the Danny presented the class the paper folding problem. The problem involved folding a strip of paper over on itself, and being able to predict the Up folds or Down folds on the strip. My partner and I used Plya's method to solve a problem.
1. Understand the problem:
We started by identifying the problem and understanding it.That being, no matter what fold we were working with, we could predict the sequence of up or down folds before the paper was folded.
2. Devising a Problem:
We discussed how we would go about solving the problem. We decided to start by folding the paper and recording the Up or Down folds each time we folded the paper and checking if we could find a pattern just by looking at it.
3. Carry out the plan:
We carried out our plan, folding the paper and writing down the Up or Down folds, indicating with a U for up or a D for down.
4. Looking Back:
We then looked at the information we gathered to search for some sort of pattern. Looking back at the data the way we had represented it there did not seem to be a pattern we could see. Because we had not solved the problem we tried a new approach.
2. Devising a Problem
We thought of a new way to go about the problem. Getting a new piece of paper we looked at a smaller case. We only looked at the first and second fold with the new paper. Examining the paper itself we concluded that on the first fold we created one U fold in the center. If we fold the paper again, we create two new folds. One Up and the other Down. The left side folded up and the second folded down. We discovered that this idea, that for each fold there would be a new fold on each side continued for the whole problem.
3. Carry out the Plan:
So we created a new list of Up or Down folds but representing it differently.

1:                                                      U
2:                                      U             U             D
3:                              U     U     D     U    U      D      D
4:                          U U D U U D D U U U D  D U  D D

4. Looking Back:
Looking at the data this way made it easier to see that earlier folds persist through the the new folds. My partner and I discovered that on each side of every second earlier fold the left side would have a new U fold and on the right, a new D fold. Writing the pattern in this "tree" pattern we could predict what the next fold would look like. Therefore solving the problem Danny proposed.