Problem Solving

FOLDING

Take a strip of paper and stretch it so that you have one fold between your left thumb and index finger and the other between our right thumb and forefinger. Fold the strip so that the left end is on top of the right end. Repeat several times, each time folding so that the left end is on top of the right end of the strip.

When you're done, keep holding the right end and unfolding the entire strip. Some of the creases point vertex up, some down. Can you predict the sequence of ups and downs for that number of times you carry out the folding operation? Can you form a convincing argument that your prediction method is correct? Can you extend the problem to two dimensions (folding first left over right, then back over front, then left over right....)?

Understand the problem:
Create a formula that can predict the sequence of ups and downs for that number of times you carry out the folding operation (for n folds).

Devise a plan:
Fold the paper and write down the folds.
Devise a formula based on n number of folds.

Carry out the plan:
Folding a piece of paper once = 2 pieces.
Twice = 4 pieces
Three times = 8 pieces
FORMULA (on n number of folds) = 2^n

Folds:
1. Down
2. Up. Down. Down.
3. U U D D U D D
4. U U D U U D D D U U D D U D D
5. U U D U U D D U U U D D U D D....(unable to continue/tell folds apart)

Look back (Conclusion):

    Left side          center        Right side
[mirror image]      Down       [previous fold]

Based on folds:
There is always an odd number of folds = there is always a centerpiece (which points Down).
The pattern represents a binary number system.

FORMULA (n = number of folds): (2^n) - 1, n>=1 (cannot have negative folds, 0 folds = 0)

Acknowledge when, and how, you're stuck:

I doubt I'd be able to build a program out of this....it would start with creating an empty list of (2^n) - 1 elements...