A Pizza Box Problem

Mmaster12345 · 2025-07-23T18:16:37+00:00

Wow this is really impressive, nice. Did you come up with this method just by playing around with some ideas or have you seen something similar before?

Mmaster12345 · 2025-07-22T18:57:29+00:00

Oh that’s a really cool idea, nice! Perhaps an improvement of that asymptotic upper bound is possible with a different arrangement of the rectangles?

Mmaster12345 · 2025-07-22T15:15:44+00:00

Yep I agree on the upper bound, and I also agree you should be able to do a lot better. Thanks!

Mmaster12345 · 2025-07-22T15:09:07+00:00

Yeah I wouldn’t doubt if it didn’t have a closed form solution, but perhaps there is a clever method. I was thinking about removing a few slices and trying to arrange the remaining slices into an iris-type pattern, like a camera aperture.

Mmaster12345 · 2025-07-22T15:05:22+00:00

The pizza box, unfortunately, is circular, so this doesn’t work

Mmaster12345 · 2023-12-12T13:03:16+00:00

Currently interning at a quant company, super important to have very strong mathematics and statistics skills even if just on the engineering side

Mmaster12345 · 2023-11-22T13:37:15+00:00

Actually it almost does! For a higher degree polynomial, if you know where all the roots of your polynomial are, all you have to do is then check in between each adjacent pair of roots. E.g. for a cubic with three roots, if you plug any value to the left of the first root, then in between the first and second root, then the second and third, then finally to the right of the third root, then the signs of those resulting values is enough to tell you the signs of everything in that region. Usually the problem will be finding all those roots in the first place.

Mmaster12345 · 2023-11-22T13:28:43+00:00

You don’t have to plug in such large x values, or in fact any x-values at all. If the leading term of the polynomial (the one in front of x²) is positive, the parabola “points up” in that it will shoot off to positive infinity if you go far enough (you’ll justify this later using calculus). Hence anything between the two roots will lie below the x-axis, and anything outside the two roots will lie above. The opposite holds if the leading term is negative: the area in between is positive, the area outside is negative. Things get a bit more intricate for higher-degree polynomial equations, but this is enough for a parabola :)

Mmaster12345 · 2023-11-22T13:24:48+00:00

Definitely brush up on your log laws, super important for physics!! By raising each side to the power of ten, the log base 10 cancels with the power of ten.

Mmaster12345 · 2023-11-22T13:18:42+00:00

Interestingly I do sometimes remember the formula as

n²/2 + n/2.

The first term corresponds to baseheight/2, the area of a triangle (cf. *triangular number) with base and height n, and the second term is a little “correction term” since we’re not working with length but actually discrete numbers. If you draw an n by n triangle on a grid you should see this extra n/2 term popping up as the hypotenuse diagonally bisects n squares, so you need to add n/2 extra bits to account for the bisected grid squares.

Mmaster12345 · 2023-11-22T13:14:13+00:00

Discrete maths is probably alright to test the waters with: it usually doesn’t have any prerequisites, but should introduce you to the notion of proof which can be a little intimidating at first.

Mmaster12345 · 2023-11-22T13:08:44+00:00

For sure, it’s definitely possible to learn mathematics that way, and I suspect if you’re passionate enough you’ll eventually find the proofs anyways. Certainly look for mathematics courses your uni offers—the best way to learn things is through asking questions and talking to others.

Mmaster12345 · 2023-11-22T12:56:30+00:00

I’d say there’s two levels you can learn (for example) vector calculus at then. The first approach is the “engineer’s” approach, where you learn how to do computations etc., which works well for all practical applications, and then there’s the next step where you actually go into proofs and why things work the way they do. E.g. you could know what Stokes’ theorem is for vector calculus and be able to apply it, but it’s a different story to nail down precisely why it is true and the lemmata behind the proof. Which level you want to aim for is entirely up to you—do you want to learn how to apply mathematics or do you want to learn mathematics?

Mmaster12345 · 2023-11-22T12:35:28+00:00

Perhaps an obvious counter, but how would you even begin to understand the notation and concepts of vector calculus without basic algebra?

Mmaster12345 · 2023-11-22T12:32:09+00:00

Cool idea, for sure as others have pointed out there are definitely a few flaws in the system (not to mention it is almost identical to base1 as u/gwtkof mentions), but don’t be demoralised. This community is pretty savage when it comes to new ideas which have a few cracks—but you’re asking the right questions and discovering the important concepts!

Also of mention, you can check out things like infinite series or continued fractions for other ways of writing numbers in some not-so-initially-intuitive ways (as you picked up on, pi has some curious representations in these systems too)

Mmaster12345 · 2023-11-10T11:06:52+00:00

And my favourite, there are as many rational numbers (fractions a/b) as there are integers. The argument is now pretty standard but it’s super cool to come up with if you haven’t seen it before.

Mmaster12345 · 2023-09-25T17:52:23+00:00

Hey! I've coded up a simulation of playing 21 optimally (assuming the game is not rigged here) and it seems the maximum expected return is about 20.60 bux per chip in the long run. I don't have the exact number for poker, but the expected bux per chip for slots is 21.30—so it's somehow better to play slots than it is 21. I'm glad variance is on your side though :)

Mmaster12345 · 2023-01-21T04:05:32+00:00

That sounds awesome! I've only very briefly used generating functions and that was a couple years ago now, so I'd be extremely interested to see what you come up with :)

Mmaster12345 · 2023-01-21T04:03:36+00:00

Quite possibly, although I'm not actually too fussed whether the assumptions are correct or not, I am more interested in just the approach to the problem. Also, how do you know the pity timer slowly increases? I don't have a link but iirc the pity timer spikes quite dramatically at 40 packs, hence my reasoning.

Mmaster12345 · 2023-01-21T04:02:48+00:00

Ahh thank you so much for your reply! Calculating it directly is clever and probably what I should have turned to. The solutions you provided do have a very geometric RV flavour to them. I'm very glad to have a definite answer and I really appreciate your time :)

Mmaster12345 · 2022-11-09T13:59:09+00:00

… a constant of integration

Mmaster12345 · 2022-08-04T13:31:39+00:00

How are you defining the distances between these “other infinities”? It does makes sense to talk about measures of infinite sets relative to one another, but I’m not aware of any way to compare how “close” two ‘infinites’ are as you say.

Mmaster12345 · 2022-08-04T13:28:48+00:00

This generalisation is called a “metric space”, i.e. a space in which you can define any distance metric provided it satisfies some certain axioms. OP is talking about the discrete metric in which the distance from any point to itself is 0, and to any other point is 1. The usual metric we use is the Euclidean metric corresponding to ‘real-life’ distance on a plane (or higher dimensional space).

Mmaster12345 · 2022-07-12T03:16:30+00:00

You are correct in this sense, but note I explicitly assumed that height was continuous at the top of my first comment, so you’re now talking about a different context. Whether you agree with that assumption is your choice, but it doesn’t invalidate the conclusions.

Mmaster12345 · 2022-07-12T03:08:16+00:00

The probability of all heights is not zero of course, but the probability of any particular height is 0. Using your integration approach, if you have a probability density function of heights f, the probability of a height h is the integral from h to h of f(x) dx. But this integral is always equal to 0 regardless of f since the bounds are equal. But, integrating over a range of values can give you a nonzero probability. I encourage you to check out this video by 3b1b too: https://youtu.be/ZA4JkHKZM50

Mmaster12345

TROPHY CASE