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chevy
chevy
Joined: Apr 15, 2011
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November 25th, 2020 at 9:39:09 AM permalink
Follow up question:

For arbitrary rectangle with Height=H, Width=W......(oriented with H>=W), What range can the area ratio (Blue/Total) of the folded result have? And for what H,W is the Blue area 50%?
DogHand
DogHand
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November 25th, 2020 at 10:22:48 AM permalink
Wiz,

Your second set disproves your earlier statement about the permissible number of evens.

Dog Hand
CrystalMath
CrystalMath
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November 25th, 2020 at 4:45:14 PM permalink
Quote: chevy

Follow up question:

For arbitrary rectangle with Height=H, Width=W......(oriented with H>=W), What range can the area ratio (Blue/Total) of the folded result have? And for what H,W is the Blue area 50%?




I get a general equation for the ratio of H/W = sqrt((ratio+1)/(3ratio -1)) .
For a ratio of 0.5, H/W = sqrt(3)

If H=W, ratio = 1 (same as folding a square in half).
If H >> W, the ratio approaches 1/3.
I heart Crystal Math.
chevy
chevy
Joined: Apr 15, 2011
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November 25th, 2020 at 7:14:55 PM permalink
Quote: CrystalMath


I get a general equation for the ratio of H/W = sqrt((ratio+1)/(3ratio -1)) .
For a ratio of 0.5, H/W = sqrt(3)

If H=W, ratio = 1 (same as folding a square in half).
If H >> W, the ratio approaches 1/3.



I agree!
gordonm888
gordonm888
Joined: Feb 18, 2015
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November 26th, 2020 at 7:40:24 AM permalink


The ratio of the area of that pentagon where the paper overlaps to the entire pentagon is 1/3



Sorry, didn't notice that the problem was a day old.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
Gialmere
Gialmere
Joined: Nov 26, 2018
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November 26th, 2020 at 11:01:07 AM permalink
Happy (US) Thanksgiving!



Turkey = ?
Cornucopia = ?
Native American = ?
Mayflower = ?
Pilgrim = ?

Have you tried 22 tonight? I said 22.
ThatDonGuy
ThatDonGuy
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November 26th, 2020 at 1:17:31 PM permalink
Quote: Gialmere


Turkey = ?
Cornucopia = ?
Native American = ?
Mayflower = ?
Pilgrim = ?



Just the numbers this time...

Turkey = 6
Cornucopia = 8
Native American = 7
Mayflower = 11
Pilgrim = 29

Wizard
Administrator
Wizard
Joined: Oct 14, 2009
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November 26th, 2020 at 4:21:01 PM permalink
Quote: chevy


***I "think" the red/blue diagram is not to scale. It makes blue appear as right angle. I "think" the line from A,D to the center of opposite side is the perpendicular.***

Call the center of the rectangle Y. Call the ends of the fold (X,Z). So side of blue triangle opposite A,D is XYZ with Y at midpoint and (A,D) to Y is perpendicular to XZ.

Area of Blue = .5 * XZ * AY = XY * AY
Area of rectangle = 2*Area of Blue + 2* Area of Red = 4
Area of Pentagon = Area of rectangle - Area of Blue = 4-Area of Blue

Ratio = Area of Blue / (4-Area of Blue)


From Trig using the rectangle diagram with XYZ added accordingly....
AD = sqrt(4^2+1^2)=sqrt(17)
AY = AD/2 = sqrt(17)/2

Call theta angle given by CAD, then
tan(theta) = DC/AC = 1/4

Theta is also angle for XAY, so
tan(theta) = XY/AY
XY = AY * tan(theta) = sqrt(17)/2 * (1/4) = sqrt(17)/8

Thus Area of Blue = XY * AY = [sqrt(17)/8] * sqrt(17)/2 = 17/16

And
ratio = Area of Blue / (4-Area of Blue)
= (17/16) / (4 - 17/16)
=17/47



I agree!

Welcome to the WoV forum, by the way.

That was a tough problem and only one member answered it correctly (you), so please consider yourself invited to the prestigious "Beer Club." This means I owe you a beer should we ever meet.
It's not whether you win or lose; it's whether or not you had a good bet.
Gialmere
Gialmere
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November 26th, 2020 at 8:02:17 PM permalink
Quote: ThatDonGuy



Just the numbers this time...

Turkey = 6
Cornucopia = 8
Native American = 7
Mayflower = 11
Pilgrim = 29


Correct!
-----------------------

Have you tried 22 tonight? I said 22.
chevy
chevy
Joined: Apr 15, 2011
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November 26th, 2020 at 10:31:01 PM permalink
Quote: Wizard



I agree!

Welcome to the WoV forum, by the way.

That was a tough problem and only one member answered it correctly (you), so please consider yourself invited to the prestigious "Beer Club." This means I owe you a beer should we ever meet.



Thanks for the "Beer Club" honor.

In fairness, CrystalMath answered my followup about the general HxW rectangle with a formula I agree with, so I assume he too had the original problem solved, just never posted.

Hopefully that doesn't preclude my membership in said club.

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