imqwerty
  • imqwerty
Fun question
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
imqwerty
  • 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.
imqwerty
  • imqwerty
@butterflydreamer
anonymous
  • anonymous
You fold it 9 times, but do you always switch directions when folding?

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imqwerty
  • imqwerty
No u can't switch directions while folding. U have to fold along the same directions.
anonymous
  • anonymous
If you fold along the same direction, then your second fold should give you 3 valleys.
anonymous
  • anonymous
Your valleys will be parallel
imqwerty
  • imqwerty
Wait I am sending a pic of valleys nd hills formed after folding the paper twice
myininaya
  • myininaya
errr my paper won't let me fold it more than 5 times :p
ganeshie8
  • ganeshie8
reminds me of Alice in wonderland
imqwerty
  • imqwerty
Lol
dan815
  • dan815
which way am i folding it in half
dan815
  • dan815
|dw:1436161935981:dw|
myininaya
  • 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...
dan815
  • dan815
whats a valley, whats a hill, im confused
myininaya
  • myininaya
the paper touches the desk=valley the paper doesn't touch the desk=hill
imqwerty
  • imqwerty
Exactly
dan815
  • dan815
ahh ok
ganeshie8
  • ganeshie8
|dw:1436162117710:dw|
anonymous
  • anonymous
I'm getting about \(2^{n-1}\) values and \(2^{n-1}-1\) hills.
myininaya
  • myininaya
my hills and valleys are becoming hard to read
anonymous
  • anonymous
\[ \begin{array}{c|c|c} \text{folds} & \text{valleys} & \text{hills} \\ \hline 1&1&0\\ 2&2&1\\ 3&4&3\\ 4&8&7\\ 5&16&15\\ 6&32&31\\ 7&64&63\\ 8&128&127\\ 9&256&255 \end{array} \]
anonymous
  • anonymous
I'm going to experiment a bit more though
myininaya
  • 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)
imqwerty
  • imqwerty
u got the correct answer @wio
myininaya
  • myininaya
my paper isn't working :p
myininaya
  • myininaya
I can't get the same thing they had for 2 folds :p
anonymous
  • 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
ganeshie8
  • ganeshie8
is the setup somehow similar to |dw:1436162688519:dw|
anonymous
  • 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 \(255 + 1 = 256\).
myininaya
  • 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 / ?
anonymous
  • anonymous
|dw:1436162935382:dw|
myininaya
  • myininaya
oh the points
myininaya
  • myininaya
and we don't consider the endpoints
ganeshie8
  • ganeshie8
i think we count only the new creases formed
imqwerty
  • imqwerty
wio is absolutely correct
anonymous
  • anonymous
First I thought it was like this: |dw:1436163027686:dw|
dan815
  • dan815
thats what im working with, is that wrong way to fold
myininaya
  • myininaya
so for 3 folds I have: |dw:1436163142668:dw|
dan815
  • dan815
you are still creating the same number of valley and hills with every fold except the first fold
myininaya
  • myininaya
oh yeah that checks out :)
ganeshie8
  • ganeshie8
|dw:1436163257855:dw|
dan815
  • dan815
ah xD im already working on the other problem though its interesting
imqwerty
  • imqwerty
1 Attachment
dan815
  • dan815
i think the solution might be the same if you fold this way too |dw:1436163499857:dw|
imqwerty
  • 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
imqwerty
  • 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
dan815
  • 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
dan815
  • dan815
|dw:1436163851891:dw|
ganeshie8
  • ganeshie8
so after n alternate folds, the paper gets dividied into 2^n parts
anonymous
  • anonymous
The points are given by horizontal and vertical folds
imqwerty
  • imqwerty
@dan815 according to the question a hill is this kind of structure- |dw:1436250404384:dw|
ganeshie8
  • ganeshie8
with alternate folds we may define hill as convex and valley as concave
imqwerty
  • imqwerty
okay but the question is not saying to do so.
imqwerty
  • imqwerty
i got some more interesting questions which are even more fun than this one.
myininaya
  • myininaya
|dw:1436164431433:dw| isn't this picture representing one hit and two valleys? it isn't just one structure right?
myininaya
  • myininaya
one hill*
imqwerty
  • imqwerty
yes the pic represents *1hill between 2 valleys
myininaya
  • myininaya
ok cool I just wanted to make sure I was happy with my understanding :)
imqwerty
  • imqwerty
( ͡° ͜ʖ ͡°)

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