The Folding Problem

The folding problem class was the best class so far in this class for me because I understood what was happening. Obviously, the reason why this class is so challenging for me is because this logic is new to me and I'm not used to thinking so logically. In time, I hope I'll understand.



Also, I apologize in advance

Problem: Folding
Given: # of folds
Required: sequence of up(u) and down(d) creases

Plan 1: (Our only plan) We would go through the folds starting from the first fold. With the new gathered info, we would try to find a pattern and figure out another plan.

Step 1: Count the sequence of ups and downs after each fold

1. d
2. ddu
3. ddudduu
4. ddudduuddduuduu

Step 2: Find the patterns
i) It was discovered that for all n ∈ number of folds, the numb er of creases is 2ⁿ -1

1. d (2¹ -1)
2. d    du (2² -1)
3. d    du    dduu (2³ -1)
4. d    du    dduu    ddduuduu (2⁴ -1)

ii) A group member also discovered that there is always one more down crease than up crease.With these two observations, predictions for the next folds

Folds	Down	Up
1	1	0
2	2	1
3	4	3
4	8	7
5	16 (25 -1) / 2	15 (25 -1) / 2 - 1

iii) Here, we were stuck because the patterns above did not predict the sequence of ups and downs. So, we tried to get more info by labelling which creases were created as a result of which fold number.

It was discovered that every new fold created new creases in a u,d,u,d order.
For example, Fold 2 created u,d
Fold 3 created u,d,u,d  or 2  of each
Fold 4 created u,d,u,d,u,d,u,d, or 4 of each
Combined, it looks like this:
                              1
                              d

              2              1               2
              d              d               u

      3      2      3      1      3      2     3
      d      d      u      d      d      u     d

  4  3  4  2  4  3  4  1  4  3  4  2  4  3  4
  d  d  u  d  d  u  u  d  d  d  u  u  d  u  u

Following this trend of putting the new number in the spaces between all the creases, the next sequence should be:
5453545254535451545354525453545
dduuddudddudduuddduuduuuddudduu

I can't figure out a way to predict the sequence of a large number of folds.

And because I can't draw this out, here's the dragon curve:
https://www.youtube.com/watch?v=NajQEiKFom4

Welcome!

Welcome to my blog! There most likely won't be any content that's more creative than all of the other SLOGS so I wish you the best of luck on your journey to try to read this boring slog!

This is the first time I have ever been exposed to logic formally and it intrigues me how counter intuitive logic can seem at first. Taking the example of the vacuous truth, it seems bizarre that if the universal quantifier describes an empty set, the second statement is always true.

It only makes sense after taking a step back and taking time to let it sink in. This is probably a sign of what's to come.