## imqwerty one year ago Fun question

1. imqwerty

U got a paper and u fold it half and then u open it and when u open it u will see a valley type of thing formed(V). If u fold the paper half twice along the same length and then open it u'll see 2valleys(V) and 1hill(^) formed. Now the question is-how many valleys will be formed if u fold the paper half 9 times.

2. imqwerty

@butterflydreamer

3. anonymous

You fold it 9 times, but do you always switch directions when folding?

4. imqwerty

No u can't switch directions while folding. U have to fold along the same directions.

5. anonymous

If you fold along the same direction, then your second fold should give you 3 valleys.

6. anonymous

7. imqwerty

Wait I am sending a pic of valleys nd hills formed after folding the paper twice

8. myininaya

errr my paper won't let me fold it more than 5 times :p

9. ganeshie8

reminds me of Alice in wonderland

10. imqwerty

Lol

11. dan815

which way am i folding it in half

12. dan815

|dw:1436161935981:dw|

13. myininaya

for 5 folds I get: 8 valleys 8 hills I think maybe we can find a pattern like you know 2,3,4 folds...

14. dan815

whats a valley, whats a hill, im confused

15. myininaya

the paper touches the desk=valley the paper doesn't touch the desk=hill

16. imqwerty

Exactly

17. dan815

ahh ok

18. ganeshie8

|dw:1436162117710:dw|

19. anonymous

I'm getting about $$2^{n-1}$$ values and $$2^{n-1}-1$$ hills.

20. myininaya

my hills and valleys are becoming hard to read

21. anonymous

$\begin{array}{c|c|c} \text{folds} & \text{valleys} & \text{hills} \\ \hline 1&1&0\\ 2&2&1\\ 3&4&3\\ 4&8&7\\ 5&16&15\\ 6&32&31\\ 7&64&63\\ 8&128&127\\ 9&256&255 \end{array}$

22. anonymous

I'm going to experiment a bit more though

23. myininaya

for 3 folds... I'm getting 2 valleys and 2 and half hills (I say two and a half because the end of my paper is standing up but the hill doesn't end because the paper does end; to me it isn't a complete hill)

24. imqwerty

u got the correct answer @wio

25. myininaya

my paper isn't working :p

26. myininaya

I can't get the same thing they had for 2 folds :p

27. anonymous

Think of a paper for which the front side is red and the back side is blue. On you first fold, you have just the red side showing up, so you create a valley On the second fold, you have a red side and a blue side on top, so you create a valley and a hill. On the third fold, you have red, blue, red, blue. This means you're creating two more values and two more hills. The way you fold it, there will always be the same number or reds and blues (except at the very start), so you'll always create the same amount hills and valleys with exception of the head start at the beginning

28. ganeshie8

is the setup somehow similar to |dw:1436162688519:dw|

29. anonymous

It's clear that after 9 folds, you'll have $$2^9$$ sections, divided by $$2^9-1$$ creases. It's clear this is an odd number of creases. Subtract it be one and divide by two to get the number of hills $$(2^9-2)/2 = 2^8-1 = 255$$. The number of valleys is just hills plus one $$255 + 1 = 256$$.

30. myininaya

hey for two folds I have this: |dw:1436162856530:dw| how should I read this like the ^ is a hill to me _ is a valley to me but what about the / ?

31. anonymous

|dw:1436162935382:dw|

32. myininaya

oh the points

33. myininaya

and we don't consider the endpoints

34. ganeshie8

i think we count only the new creases formed

35. imqwerty

wio is absolutely correct

36. anonymous

First I thought it was like this: |dw:1436163027686:dw|

37. dan815

thats what im working with, is that wrong way to fold

38. myininaya

so for 3 folds I have: |dw:1436163142668:dw|

39. dan815

you are still creating the same number of valley and hills with every fold except the first fold

40. myininaya

oh yeah that checks out :)

41. ganeshie8

|dw:1436163257855:dw|

42. dan815

ah xD im already working on the other problem though its interesting

43. imqwerty

44. dan815

i think the solution might be the same if you fold this way too |dw:1436163499857:dw|

45. imqwerty

ok 1 fold-1valley 2 folds-2 valleys 1hill 3folds - 4vallyes 3hill 4folds- 8valleys 7hills we observe that always the no. of valleys is one more than the no. of hill and also that the sum of valleys+hills=2^(no. of folds) -1 so with this info we can find the no. of valleys after 9 folds

46. imqwerty

u can fold the paper along the x axis or along the y axis but u cant flip directions because then hills and valleys won't form

47. dan815

they still do form, but you have to define what is a hill because u can have the same hill span through multiple points, but if u define a hill or valley to be inbetween the intsersections of the folds

48. dan815

|dw:1436163851891:dw|

49. ganeshie8

so after n alternate folds, the paper gets dividied into 2^n parts

50. anonymous

The points are given by horizontal and vertical folds

51. imqwerty

@dan815 according to the question a hill is this kind of structure- |dw:1436250404384:dw|

52. ganeshie8

with alternate folds we may define hill as convex and valley as concave

53. imqwerty

okay but the question is not saying to do so.

54. imqwerty

i got some more interesting questions which are even more fun than this one.

55. myininaya

|dw:1436164431433:dw| isn't this picture representing one hit and two valleys? it isn't just one structure right?

56. myininaya

one hill*

57. imqwerty

yes the pic represents *1hill between 2 valleys

58. myininaya

ok cool I just wanted to make sure I was happy with my understanding :)

59. imqwerty

( ͡° ͜ʖ ͡°)