Crazy UP and DOWN

         Crazy “up” and “down”? Crazy “up” and “down”!

Have you ever thought about making a paper into several ups and downs by folding it in a certain way and then record the result for each fold to create an algorithm for it? I always feel this kind of questions should be put on mathematic competition paper. However, we are actually doing this on our computer science class, which make me, a math girl, feel excited so much!

  Before getting answer of this question, there are several steps we can follow. We need to follow the instructions first, and then write down each step’s result, next we need to analyze those records, and finally we can get the answer.

(Understand the question and follow the instructions)

We were told to fold the paper in a certain way. This is really important; because you need to follow the rules and then we can get the correct results. As we did in class, some people fold in a different way, which means that they did not follow the instructions so much; thereby they got a completely different answer. And our instruction is that we fold the paper from left-hand side to the right-hand side, and record the direction of the crease, and then fold the paper again from the left (closed side) to right (open side).Keep doing this folding, and write down every result, we can hopefully find a pattern of all those results. However, it’s impossible to fold the paper as many times as we want, because as we fold the paper for six or seven times, the folded paper now is too thick that human cannot have such strength to fold it anymore. So the only thing we can do is to analyze the results we already get so far and estimate the following ones, which means we need to create an algorithm for the patter.

(Pattern I tried to find and some sample pattern)

The one I tried:

As we can see from above pictures, we notice that as we do one more fold, we will one “up” and one “down” just beside new ceases we got from the previous fold. So the new creases are actually based on those previous ones. So if we can build up an algorithm in DrRacket, to give this kind of define, then we can get the patterns right away just like we did on the Fibonacci one. And then we put every time’s creases “up” and “down” together, so we will get the final answer for the question.

Sample answers:

1.  As we were shown in Tuesday’s lecture, we can draw the creases and after we do some samples, we can easily find that if we draw a line as the symmetrical axis, we can rotate the right-hand part counter-clockwise for 90°. And then if we want to get the next one, we just need to copy the previous one’s shape on the right side, and then rotate it for 90°to get the left side. And we will get the new crease. So in the same way, we also need to use the previous one to get the next one.

2.  For the second one, we write down each fold’s creases. ( "U" means "UP", "D" means "DOWN".)

①                                                                D

②                                                             UDD

③                                                        UUDDUDD

④                                             UUDUUDDDUUDDUDD

⑤                       UUDUUDDUUUDDUDDDUUDUUDDDUUDDUDD

⑥UUDUUDDUUUDDUDDUUUDUUDDDUUDDUDDDUUDUUDDUUUDDUDDDUUDUUDDDUUDDUDD

So from the above records, we can find that the right side of each fold is exactly the previous one, and except for the middle D, we also notice that that each “up” and “down” is just opposite to the one it faces on the right side. So the same theory, we still need the previous one to get our next pattern of the creases. And if we want to do maybe the 1000^th fold, it is really a huge work to do. But as long as we follow the ways we found from the above three methods, we can get it, even though we need enormous amount of time!