Problem solving episode: paper folding

Problem:

When a strip of paper is continuously folded left-end over right, determine the relation between the number of folds and the number of creases and the position of the creases.

ie. When the paper is folded 8 times, what are the position of the creases?

Data: 

0 Folds = no crease 
1 fold = 1 crease (D) 
2 Folds = 3 creases (U D D) 
3 Folds = 7 creases (U U D D U D D) 
4 Folds = 15 creases (U U D U U D D D U U D D U D D)

Plan: 

Try to find the pattern to how the position of the crease relate to the number of creases and also make connection between number or creases and number of folds.

Solving:

 

Step 1: look back at the data and see if anything obvious pops up

 F= 0 c = 0

 F= 1 c = 1

F= 2 c = 3

 F= 3 c = 7 

 F= 4 c = 15 

I can derive a formula such that number of crease = 2( number of crease (number of folds - 1)) + 1

 

say for example, folds = 3

# crease = 2( number of crease ( 3-1) )+1 =  2 ( 3 ) + 1 = 7

thats consistent with our data!

 

Step 2: try to figure out the position of crease

 

looking back at data:

1 ) (D)

2) (U D D)

3) (U U D D U D D)

4) (U U D U U D D D U U D D U D D)

 

- it looks like the pattern is that 

 

1 ) (D)

2) (U D D)

3) (U U D D U D D)

                                                  4) (U U D U U D D D U U D D U D D) 

we can simply say that from 1 to 2, we add U to the left of D(bolded) and D to the right of D(bolded)

from 2 to 3, we add U to left of U(red) and D to right of U(red). and then we do the same to the red D on the right. -> adding U to left and D to right to each.

 

 To find the position in the proceeding steps, one adds a U to the left and a D to the right of the positions in the present step while ignoring positions that have previously had a U added to their left and a D to their right. 

Done :)