logic and paper == brain with paper cuts!

During week 3 of CSC165, we were given a deceptively simple problem to tackle and solve. The problem given to us was...

Take a strip of paper and stretch it so that you have one end between your left thumb and index finger and the other between your right thumb and forefinger. Fold the strip so that the left end is on top of the right end. Repeat this several times, each time folding so that the left end is on top of the right end of the strip. 

When you're done, keep holding the right end and unfold the entire strip. Some of the creases point vertex up, some down. Can you predict the sequence of ups and downs for any number of times you carry out the folding operation? 

Being the recorder (the man who wrote down on all thought, hypotheses and predictions about the problem), I am in a good position to apply G. Polya's problem solving approach to this problem.
Hence, this post will follow a specific structure:

1. I will understand the problem.
2. I will devise a plan.
3. I will carry out the plan.
4. Look back and reflect.
5. Acknowledge when and how I am stuck. 

Let's begin!

Needing some material to work on and evaluate my results, I made the first step of my plan to be:

play and fold the paper.

The other steps of my plan are the following.

1. Play and fold the paper, observe the patterns of the folds.
2. Understand how the variable, number of folds effects the patterns
3. Make a table of the number of folds vs. the pattern generated.
4. Using the observable pattern, try and predict the sequence of folds.

Here is a video of me, folding the paper:

I was only able to fold the paper four times as the paper got thicker and harder to work with...

Nonetheless, I got substantial data.

Num. folds	pattern	pictogram
1	D	$\vee$
2	UDD	$\wedge\vee\vee$
3	UUDDUDD	$\wedge\wedge\vee\vee\wedge\vee\vee$
4	DDUDDUUDDDUUDUU	$\vee\vee\wedge\vee\vee\wedge\wedge\vee\vee\vee\wedge\wedge\vee\wedge\wedge$
5	...	...

Here a 'D' represents a down fold, like a $\vee$ and a 'U' represents an up fold, like a $\wedge$.

I found it very hard to understand the pattern at first, but as I kept searching for any repetition, inverses in the sequence, I realised 3 things.

1.  The pattern builds around a central D, (highlighted in red)
2. Every extra fold adds $2^{n}$ folds, where $n$ is the initial number of folds in the paper.
3. The pattern reverses and inverses around the central D.

Using these properties, I could predict the next sequence! This can be done by adding a sequence of D's and U's in between the already existing sequence.

Confusing? Well, yes. But it is best shown by the illustration below:

Every new sequence just builds on another sequence, like a tree

I am so happy to have cracked this out...