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Mountains and Valleys (Posted on 2006-07-19) Difficulty: 2 of 5
In origami there are two ways of folding a piece of paper. The usual way is to fold the paper upwards and if unfolded the crease is called a valley fold and from the side it looks like this: \/ If the paper is folded backward instead the crease is called a mountain fold and looks like this: /\.

If you take a strip of paper and fold it in half twice (left over right) and then unfold it without flipping you will get a figure that looks something like this from the side:

/\_/

Which consists of the following sequence of folds:

/\ \/ \/

Describe the sequence of folds if the original strip of paper had been folded over 10 times and then unfolded.

No Solution Yet Submitted by Jer    
Rating: 2.5000 (2 votes)

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Solution Got it! Comment 9 of 9 |
It just hit me!  The structure of these massive strings are fractal in nature.  Look at the mountains and valleys introduced with each fold (Using the F(4)=AAVAAVVVAAVVAVV string):
Original  AAVAAVVVAAVVAVV
Fold 1           V
Fold 2       A       V
Fold 3     A   V   A   V
Fold 4    A V A V A V A V
Each successive fold interleaves a AVAV....AV sequence into the previous sequence! For example:
4 Folds     A A V A A V V V A A V V A V V
AV String  A V A V A V A V A V A V A V A V
5 Folds    AAVAAVVAAAVVAVVVAAVAAVVVAAVVAVV
Counting from the right, the mountain folds are at positions 3,6,7,11,12,14,15,19,22,23,24,27,28,30,31,... and the valley positions are at 1,2,4,5,8,9,10,13,16,17,18,20,21,25,26,29,... Because the right half never changes this is true for all strings.

More concretely the sequence of mountain folds is numbers expressible in the form 2^n*(4k+3) OEIS A091067 and the valley folds are its complement numbers expressible in the form 2^n*(4k+1) OEIS A091072

In my previous post I mentioned the binary expansions of a formula, that binary expansion going right to left is OEIS A038189, starting from the second term.  From that sequence, there are references to other effectively identical sequences including ones named "The regular paper-folding (or dragon curve) sequence" A014709 A014710 A014577 

Edited on March 5, 2016, 11:30 am
  Posted by Brian Smith on 2016-03-05 11:23:03

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