Third Week: Answers to the Exciting Folding Problem

This class has probably been my most favorite class of all, probably because I am such a big fan of logic problems. And although the folding problem seemed pretty challenging at first, after giving it a little thought and using observation, I was able to find the pattern very quickly, before the hints were given. So below are the answers to this problem explained thoroughly, I hope.

Obviously the first step was to test the first few cases, for which I got the following results ('D' is down crest, 'U' is the up crest):

Fold #1  =            D
Fold #2  =         U D D
Fold #3  =  U U D D U D D

The first major observation that helped me crack the problem was to notice that the middle element is always 'D'. Then, it was just a matter of identifying what comes after that 'D' and before that middle 'D'. To get the answer, we have to realign the answers to the right, with the middle D underlined:

Fold #1  =                                             D
Fold #2  =                                       U D D
Fold #3  =                           U U D D U D D
Fold #4  =  U U D U U D D D U U D D U D D

Do you see it now? The pattern on the right of the middle 'D' is the pattern of the previous fold. Now, it's just the matter of identifying what is the pattern on the left of the middle 'D'. But if you look at the first patterns (aligned center), and compare the results, you'll see that they are symmetrical in a way. If you switch every 'U' for 'D', and 'D' for 'U' on the left side, you'll see the symmetry:

Fold #2  =         U D D                               =               D D D
Fold #3  =   U U D D U D D                         =         D D U D U D D

Therefore, the left pattern is the reverse of the opposite values of  the right pattern. Mathematically, I guess you can represent the answer the following way:

New-Pattern  =   reverse (not Old-Pattern) + 'D' + Old-Pattern


Geometrically, we can also see a pattern, as shown below: in the red we see the previous pattern repeated, and in the dashed green we see the old pattern again, only rotated by 90 degrees.