Program:Take a strip of paper and stretch it so that you have one end between your left thumb and index finger and the other between your right thumb and fore finger. Fold the strip so that the left end is on top of the right end. Repeat this 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 unfold the entire strip. Some of the creases point vertex up, some down. Can you predict the sequence of ups and downs for any number of times you carry out the folding operation?
Understand the problem
I know that I can take a piece of paper, and stretch it and hold it using both hands. Then I fold the paper so the left end is on top of the right end. After I unfold it,there is a crease point down, we call it "down". We refold it like we did before, then fold it again (time 2), after unfolding, the pattern is up, down, up. Now we know how to fold the paper properly.
I know that I can take a piece of paper, and stretch it and hold it using both hands. Then I fold the paper so the left end is on top of the right end. After I unfold it,there is a crease point down, we call it "down". We refold it like we did before, then fold it again (time 2), after unfolding, the pattern is up, down, up. Now we know how to fold the paper properly.
The question is clearly asking for a method to predict the crease pattern, say, at time 5, what pattern of "up" and "down" folds do you get?
- Devise a Plan
- Plan A: write down the pattern and observe if there is any pattern.
- Carry out the Plan
- Plan A: write down the pattern
- time 1: d
- time 2: u d d
- time 3: u u d d u d d
- time 4: uuduuddduuddudd
- time 5: uuduudduudduddduuduuddduuddudd
- conclusion: it is pretty hard to see any pattern.
- Devise another Plan then...
- Plan B:
- visualize the problem, draw pictures.
- Carry out the Plan:
- Observation and Conclusion: After drawing what pattern looks like every time we fold the paper, We draw a red straight line through it center.We can see that the left half of the picture is the previous picture, and the right half is a rotated- 90-degrees-counterclockwise version of the left half. The blue line indicates this pattern.The center crease is always "down".
- After visualizing and find the pattern, we go back to the plan A to confirm it.
- Plan A: write down the pattern
- time 1: d
- time 2: u d d
- time 3: uud d udd
- time 4: uuduudd d uuddudd
- time 5: uuduudduuddudd d uuduuddduuddudd
- Pattern:
- the center is always d. left part is the copy of the previous one's pattern. The right part is symmetric to the left part with respect to the center point.
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