Understanding The Problem:
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At first glance, I assumed this was going to be easy, since it's titled "Folding", like honestly how hard could it possibly be right? It's "Folding"!... so much for that assumption...
The problem is to take a strip of paper and fold it several times, while analyzing the pattern of down(s) and up(s) that occur upon each fold. This is restricted to a consistent orientation. The objective of this problem is to come up with an output being a formula/method in predicting the sequences of up(s) and down(s) given a number of folds.
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Devise A Plan:
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Nevertheless, after reading the handout I thought it would be a wise move to follow the instructions provided on the handout And so, I devised the following plan.
[Step 1] Begin folding strips to see what occurs and understand what is meant by the problem practically.
[Step 2] Restart with a fresh strip and begin recording the number of ups and downs to see if there is a numerical pattern
[Step 3] If numerical pattern does not seem to exist or no method/formula can be concluded, restart from the beginning and record the actual pattern displayed between each folding. (1-5 folds should be sufficient)
[Step 4] Analyze the illustrated pattern from step 3
[Step 5] If no eureka occurs, go to office hours :)
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Carry Out The Plan & Look Back:
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[During Step 1] I was instantly intrigued by this problem, it presented itself to be much more difficult than I had realized initially. No pattern seemed to present itself during this stage of mass fiddling.
[During Step 2] After recording the number of up(s) and down(s), which was extremely terrible at fold 4 and 5. I managed to conjure up a pattern, being that the # of down(s) increases by a multiplication of 2 each fold (this made me feel pretty good, since it was some progress), initially starting a 1 and that the # of up(s) held the following increasing patterns 0, 1, 2, 4, 8... initially starting at 0. I soon realized that this wasn't much of any help at all and proceeded with step 3.
[During Step 3] Recording was definitely annoying at fold 4 and 5 once again. It was like reading the cheat sheet I made for test 1... (....Okay, no not really, but it was definitely tedious). The reason was that it became much more difficult to read as the up(s) and down(s) were becoming less defined upon fold 5. Nevertheless as planned, I had proceeded to analyze the data. It took approximately 3 minutes before I had realized that the previous pattern of up(s) & down(s) were preserved upon the right of the next pattern to follow.
Ex/ ↑↓
1-Fold: ↓
2-Fold: ↑↓↓
3-Fold: ↑↑↓↓↑↓↓
4-Fold: ↑↑↓↑↑↓↓↓↑↑↓↓↑↓↓
5-Fold: ↑↑↓↑↑↓↓↑↑↑↓↓↑↓↓↓↑↑↓↑↑↓↓↓↑↑↓↓↑↓↓
This gave me a strong glimpse of hope that I may be able to solve this problem soon!... Unfortunately about an hour had passed and I still didn't notice any other patterns besides that the extremes were of opposite directions consistently for each fold. It wasn't til 10 minutes after this that I had found a mirrored pattern marching to the center of each fold and within the center lied the FIRST FOLD!. This was the eureka moment I had been waiting for! It felt tremendously awesome
Illustration:
1-Fold: ..................................↓
2-Fold: ...............................↑ ↓ ↓
3-Fold: .......................... ↑↑↓ ↓ ↑↓↓
4-Fold: ..................↑↑↓↑↑↓↓ ↓ ↑↑↓↓↑↓↓
5-Fold: ↑↑↓↑↑↓↓↑↑↑↓↓↑↓↓ ↓ ↑↑↓↑↑↓↓↓↑↑↓↓↑↓↓
Hence, the solution to predicting the sequence of up(s) and down(s) for the n-th fold is to know the previous sequence and maintain a mirrored version of that previous sequence on the left and right of the first orientated fold (as shown above).
However, this is all assumed relatively like induction, as it works for 1, 2, 3, 4, 5 folds as the base case, it may not work for the 6th. Then comes the reasoning behind the mirrored series' appearance, which is the folding motion of say... the left end to the right end (or the original orientation used). This flips the previous folds held within the left section, but maintains the center being the 1-Fold, this implies that the (n + 1)-th fold will maintain the same structure/pattern.
Ex/ ____o____ ........................../
.........................................../ ....↓
........................................./
.......................................o____________
Therefore, Q.E.D :) !
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Acknowledge When, And How, You're Stuck:
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[During Step 2]: This step was a mistake since I was working with the # of up(s) and down(s), not the sequence... unfortunately I didn't realize this until the end.
[During Step 3]: My 2nd mistake I believe was within that approximately 1 hr duration period, which I was stuck. What I should have done was focus on what was occurring to the folds that had already been made during the next fold to come. This probably could have led me to the final answer more quickly, but instead I focused too much on deriving a pattern within the illustrated example.
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