A unit square is cut into rectangles. Each...

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You are currently reading a thread in /sci/ - Science & Math

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A unit square is cut into rectangles. Each of them is coloured by either yellow or blue and inside

it a number is written. If the color of the rectangle is blue then its number is equal to rectangle’s

width divided by its height. If the color is yellow, the number is rectangle’s height divided by

its width. Let x be the sum of the numbers in all rectangles. Assuming the blue area is equal

to the yellow one, what is the smallest possible x?

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>>7757943

>2 colors

>solid distribution

>.jpg

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>>7757945

what?

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>>7757975

The only way to do that would be to divide the square into infinatly many horizontal rectangles painted blue

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>>7757943

I'm starting to think you can't go below x=2.5 but i'm probably wrong

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>>7758026

probably this

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>>7758048

How do you reach 1?

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>>7758063

Don't cut the square into more than one rectangle, I think. Regardless syntactically the question states that it's cut into rectangles, which I think automatically rules that answer out.

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>>7758109

here's (roughly) your proof.

each blue rectangle has a minimum value of 1. There can be no blue rectangles, if cut into more than 1 shape.

play around with all-yellow thing.

Only one problem though.

> Assuming the blue area is equal

to the yellow one

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>>7758024

integrate over, equals to 1, but still wrong.

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This makes me think the minimum is 2.5

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>>7758242

I tried lots of similar reasonings and i can't get below 2.5 either, does OP have the definitive answer to this problem?

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>>7758314

dont mind me im retarded

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>>7758314

True

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>>7758346

>assuming the blue area is equal to the yellow one

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>>7758348

True

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not sure if this changes anything, but both yellow and blue areas are 0.5.

just have to prove minimum number for yellow area is 0.5 and for blue is 2.

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>>7758386

Yeah that's what I've been working with

Gonna go for now

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>>7758242

how about this

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>>7758314

> cut it in n equal yellow rectangles, n > 0

> n * 1/n = 1

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>>7758435

this way doesn't seem able to get me under 2.5 either, because i can't have an infinite amount of subrectangles right?

I mean if we're allowed to have infinite subrectangles then the answer is 0

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>>7758465

No, cutting it up into an infinite amount of rectangles will make x approach infinity, even if the h/w or w/h of the rectangles approach zero.

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>>7758435

Rectangle width doesn't have to decrease linearly; this anon is correct as well >>7758476

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>>7758435

Let m be the ratio between the area of the interior square and the area of the exterior square

x = sum(k=0)^inf 4(sqrt(m^k)-sqrt(m^(k+1)))/(sqrt(m^k)+sqrt(m^(k+1)))

x = sum(k=0)^inf 8/(sqrt(m)+1)-4

Interestingly enough the values for the 4 rectangles that make up each interior square don't change no matter how small the square gets. So x is infinite.

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>>7758486

ofcourse, but not if you also decrease the width of all the blue rectangles and length of all the yellow rectangles.

define w as the width of the blue and the length of the yellow rectangles. Let's look at just the first "ring" of 4 rectangles. as w approaches 0, x approaches 4*w = 0. the square left inside is just the unit square scaled by some constant very close to one, so by induction and mathematical wizardry we can say x=0.

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>>7758592

I suggest you actually look at the infinite sum of the ratios. Even as the widths of the blue rectangles and heights of the yellow approach zero, the sum approaches infinity. You can't just calculate the sum of an infinite amount of ratios that each approach zero, it only exists as a limit.

See >>7758508

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Alright, I'm gonna just lay out what I've done so others can take over.

A) We know the blue and yellow shades must cover spaces of 0.5 x 1 exactly. We must prove minimum values for both yellow and blue shades.

B) So let's prove that this must mean that if we divide the yellow rectangle into pieces with height <= 0.5 and width <= 1, the minimum value for x in yellow is 0.5:

If one of the subrectangles has height >= width/2, your ratio is obviously greater than 0.5. This case cannot work as you can already have a smaller division (0.5 x 1.0).

If no subrectangle has a ratio of greater than or equal to 0.5, it means that every division of the left side is smaller than half of every division of the right side. <insert proof of the number of every subrectangle here totalling a number >= 0.5>

C) Similarly the proof works out for the blue side as well.

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>>7758592

I will explain in detail how it is calculated so you can see how your mathematical wizardy fails.

Let me be the area of the square created by the ring of rectangles. Then the sides of this square are sqrt(m). So to find the height of one rectangle we subtract this from the height of the unit square and divide by 2.

h = (1-sqrt(m))/2

And that must mean that the width of this rectangle is the difference between the width of the unit square and the width of the rectangle it touches. Since all four rectangles are the same shape, the width of that rectangle is equal to the height of this rectangle.

w = 1-(1-sqrt(m))/2 = (1+sqrt(m))/2

h/w = (1-sqrt(m))/2 / (1+sqrt(m))/2 = (1-sqrt(m)) / (1+sqrt(m))

Now let's look at the new interior square. This is identical to the unit square except scaled down by a factor of m, forming a new interior square of area m^2. So we can use the same logic to find

h = (sqrt(m)-sqrt(m^2))/2

w = sqrt(m)-(sqrt(m)-sqrt(m^2))/2 = (sqrt(m)+sqrt(m^2))/2

h/w = (sqrt(m)-sqrt(m^2)) / (sqrt(m)+sqrt(m^2))

hopefully by now you can see that the sum of the ratios of the 4 rectangles in the kth interior square will be

(sqrt(m^k)-sqrt(m^(k+1))) / (sqrt(m^k)+sqrt(m^(k+1)))

But this simplifies very nicely to 8/(sqrt(m)+1)-4

In other words, no matter which interior square we are in, the sum of the ratios will be the same. And since there are infinite interior squares, the sum of all the ratios approaches infinity even as m approaches 1.

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>>7758670

that makes a lot of sense. thanks!

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>>7757962

You heard me. You jpegged your file, anon!

Why'd ya jpeg your file, anon? Why'd ya do it?!

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>>7758024

Divide half the square into blue triangles with width 0. Divide the other half of the square into into infinite yellow rectangles with height 0.

Therefore the minimum value for x is 0

Maximum value is infinite

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>>7759060

Let me elaborate on that poorly written post:

Divide the square horizontally. Top half of the square is cut into infinite blue rectangles with width 0. Bottom half is cut into infinite yellow rectangles with height 0. Blue area = yellow area, and x = 0

Something tells me one of you guys is going to school me and that I missed something really obvious cos the answer can't be that easy

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The answer is 2.

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>>7759285

How?

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>>7757943

Does anyone proved this?

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>>7761599

the dimension along a given axis

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>>7757943

0

assuming width and height can be assigned arbitrarily

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>>7759071

Your taking 0 times infinity to be zero. You can't do this. You must examine the limit of x as the number of rectangles approaches infinity.

This goes for everyone who says the minimum is 0.

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