Problem solving episode

So at the end of the last lecture we worked on figuring out if folding a piece of paper over and over again in the same way produces a pattern that is mathematically predictable or not. This is how I broke the problem down according to Poyla's method of problem solving.

1.  We know that we must take a strip of paper and fold its left end over to its right end several times to get a creased pattern on the strip when it is opened up. Each crease thus created is either pointing vertex up or vertex down.  The goal of the problem is to determine whether one can logically predict which way the creases will face(up or down) regardless of the number of times the paper is folded over.

2. The plan is to first fold the paper different amount of times and note the pattern of creases created for different amount of folds. Then identify the similarities and differences in a series of patterns in order to predict the direction of the creases after each fold.

3. Folding the paper once gives:                                    v
Folding it twice gives:                                                   ^ v v
Folding it thrice gives:                                            ^ ^ v v ^ v v
Folding it four times gives:                          ^ ^ v ^ ^ v v v ^ ^ v v ^ v v
and so on.

4. Looking at the pattern created after each fold we can see how it follows from the previous fold. For example, after the third fold, the paper has the same creases as after the second fold, and then some more. To make it clearer we depict it like this:

v

^

v

v

^

^

v

v

^

v

v

^

^

v

^

^

v

v

v

^

^

v

v

^

v

v

After putting it like this one can see that once a crease is made after a fold, it is (linearly) preceded by a vertex up crease and followed by a vertex down crease in the next level of folding. Also, only a new crease (i.e. a crease that was not existent on the previous level of folding) is preceded and followed in that manner. An old crease(one that was created before the previous level of folding) will just stay the same with no additional creases around it except for those that are created as a result of the other new creases.

5. You will get stuck in solving this problem if you jump straight into it with no plan of action. It takes a while to wrap your brain around the fact that the creases are not in fact random and do depend on the way you fold them, but once you try folding the paper a few different amount of times, you will see the pattern. Writing out what you see really helps in understanding what's happening.