Last poster before threadlock gets a cookie (cookie thread (Part 7)) (Part 8)

[ElizaThePsycho (Willow)]

yeah

[Zugzwang]

@May

Graphical proof is ugly and boring so calculus

Volume of a cube is lwh
The number of blocks needed to build the frame is 4(l+w+h)=k
Let’s just say k=4

V=wh(1-w-h)=wh-w^2h-wh^2

To find extrema, take partial derivatives with respect to each and set to zero

dV/dw=h-2wh-h^2=0
dV/dh=w-w^2-2wh=0
w=1/3 h=1/3 therefore l=1/3

Generally maximal volume is always a cube so you can cheat lots of optimisation problems this way

NYAAAAAAAAAAAAAAAAA calculus

[Zugzwang]

@Zugzwang

NYAAAAAAAAAAAAAAAAA calculus

calculus is fine but it’s the boring way to do the problem tbthtbhtbhtbthbtbthbt
removes the JOY and WHIMSY from solving them without it

[TodaysStory]

ok so
i get all of the individual parts. They make lots of sense in my brain. Those are things that compute. They do not fit together
I think this is something I need to just slot into my pondering time because i just dont click sometimes. like my mermeory

[May]

If you increase the starting perimeter then the l w h just change proportionally. I think the clearer thing to do would be to set k=1 so you learn all the dimensions as fractions of your total number of blocks

[May]

What joyful and whimsical method would you recommend? Number theory?

[May]

It’s a calc problem of course I’m gonna do calc

[benguinedparbecue]

@Atlas

0.9999999999999999(etc) doesn’t really equal 1

[TodaysStory]

i am at 7x10x10

[TodaysStory]

i. uh. dont actually know what to do with that information now that i think about it

[Marshal]

i spend all week being forced to do math and am excited to go to fol on the weekends and talk to people

everyone else on this forum spent the whole week being forced to talk to people and go to fol on the weekends to do math

[May]

Oh you have no fucking idea how true this is

[May]

The lore is Bad

[Zugzwang]

I think an argument without calculus should be

Suppose the optimal solution was not a cube. Then the length and width (or height, but we can assume it’s not the height without loss of generality) are not equal. Again without loss of generality, we can assume the length is lower than the width.

Increasing the length by x and decreasing the width by x leaves the overall number of blocks of the frame unchanged, but has increased your volume, since (length + x)(width - x) - length * width = x*width - x*length - x^2 = x(width - length - x) > 0 as long as x < (width - length).

It occurs to me now that if you are restricted to integers then maybe the optimal solution can’t be a cube though thbthbthtbhbhtbt

[May]

Okay fair enough

[Zugzwang]

@Zugzwang

I think an argument without calculus should be

Suppose the optimal solution was not a cube. Then the length and width (or height, but we can assume it’s not the height without loss of generality) are not equal. Again without loss of generality, we can assume the length if lower than the width.

Increasing the length by x and decreasing the width by x leaves the overall number of blocks of the frame unchanged, but has increased your volume, since (length + x)(width - x) - length * width = x width - x length - x^2 = x(width - length - x) > 0 as long as x < (width - length).

It occurs to me now that if you are restricted to integers then maybe the optimal solution can’t be a cube though thbthbthtbhbhtbt

this sounds convoluted but as long as you know the fact that the square is the best way to optimize area, then it is fairly intuitive

[notblackorwhite]

@Atlas

whilst 0.9999999999999999(etc) doesn’t really equal 1, mathematicians say it does because they’re a bunch of nerds who don’t want to redo their work when they realise they made an error (this comes up a lot)

0.9999… absolutely is exactly equal to 1. Its fraction is 3/3.

[May]

I also essentially did that but I always forget you can just make that argument

[benguinedparbecue]

it really does equal 1 but every argument you learn for why before college sucks

[May]

Like when she asked the question I was like “yeah it’s just a cube” using this intuition but I’m so used to doing this exact problem and not being allowed to do that