Folding

Unknown: the sequence of ups and downs by folding a strip of paper
Condition: folding left over right
Data: need to be collected by our own

 

Plan A:

Step 1: Fold a strip of paper
Step 2: Record the ups and downs
Step 3: Repeat the process for different number of folds
Step 3: Find the patterns

 

Plan B:

Step 1: Same as Plan A
Step 2: Find out why it is producing up/down by looking at the paper (folds) and the way we folded it
Step 3: Find the sequence based on the reason discovered in Step 2

 

We started by carrying out Plan B first. Unfortunately, as the folds increase, we lost track of the ups and downs. And it was really difficult to find out the answer by simply looking at the folds.

As a result, we changed our plan.

Here was the data that we collected.

 


By looking at the total number of the ups and downs, I realized that the ups and downs might be somehow symmetrical.

The total number of ups and downs was related to the previous one.

Let c = the current total number of ups and downs, p = the previous total number of ups and downs
We found out that c = p*2 + 1, and c must be an integer that is greater to 1.
Therefore, I believed the ups and downs were somehow related to the previous ups and downs.

Also, based on the hint given by the professor, we founded out that our first fold were always in the middle of the overall pattern of folds. Because of this, the ups and downs must be symmetrical.

Therefore, I made the following table to figure it out.

 

Fold

1

D

2

U

D

D

3

U

U

D

D

U

D

D

4

U

U

D

U

U

D

D

D

U

U

D

D

U

D

D

As you can see, the first fold always occurred in the middle.

We also found out:
1.       The ups and downs on the right side of the first fold were always the copy of the previous ups and downs.
2.       The ups and downs on the right were flipped to produce the ups and downs on the left side, which means the ups and downs were mirror symmetric.

This can be explained by the way of folding the paper. The U’s and D’s on the left side were the same as the right side before we unfolded the paper for the last time. If we unfolded the paper, these U’s and D’s would be flipped, which was exactly what we founded out on above.

According to the explanation above, I realized that this flipping-process was being repeated each time when we unfolded, and the center one was always D.

According to this discover, I made my predication for 5 folds.
If you unfold for the first time, there will be only one D.
Unfolding 2^nd time: UDD
Unfolding 3^rd time: UUDDUDD
Unfolding 4^th time: UUDUUDDDUUDDUDD
Unfolding the last time: UUDUUDDUUUDDUDDDUUDUUDDDUUDDUDD

To make life easier, I made a Python function to predict the sequence given the number of folds.
 

 

 

 

To examine the conclusion, we decided to make 5 folds.

It was UUDUUDDUUUDDUDDDUUDUUDDDUUDDUDD, which was exactly what we predicted.

And there were 31 U’s and D’s in total; we could get the same answer by using the formula c = p*2 + 1.

Q.E.D.