Homogeneous Polynomial Puns

I tried asking this on Stack Exchange but didn't get an answer, so I thought I'd try here. Here's my question:

http://mathb.in/19009

No spring noise, zero friction, cubes faster than ever! This year we got the first mass-produced maglev models of cubes, and their promises. It's still a fresh technology (for cubing, at least) and there are still some misconceptions about it going around, so let's take a closer look at how maglev actually works and understand what it does that's different from springs, and learn about tensioning systems along the way.

Intro and Terminology

I bet pretty much everyone here is familiar with cube terminology, but I want to make sure we're starting with a level playing field. (Feel free to skip this section.)

The screws that hold a cube's center pieces in place aren't screwed in all the way, but rather there's some room for the center pieces to move outwards. This amount of "room" is adjustable by turning the screw, and we call it the screw depth, or axis travel distance, or just travel distance (or other names, even). When a cube has a lot of travel distance (= a lot of room for the pieces to move outwards), it will feel more flexible and faster, but also less stable and may pop or corner twist. On the other hand, decreasing travel distance makes the cube less flexible but also more stable.

But each center piece also has a spring inside it so that the pieces don't jiggle around freely. The spring sits between the bottom of the center piece and the screw head, providing a force that pushes the pieces toward the core. You can use springs of different strengths to get different qualities without having to change the travel distance: Stronger springs make the cube more stable but also more rigid, while weaker springs make the cube looser and more flexible but also less stable. That's helpful when you want to change how stable/flexible the cube is, without changing the other properties that come with turning the screw, like the corner cutting or the cube's propensity to pop.

In the past, changing springs used to be hard and few people ever did it, but then Gan cubes started to come with easily swappable springs, and nowadays every flagship cube comes with a system that lets you change the stiffness of the spring without swapping it at all by simply applying some compression to the spring. That's called a spring tensioning system, and it lets you apply some pre-compression to the spring, which is equivalent to using a spring that was stronger to begin with. (*That's not exactly true, as we'll see.)

Compression curves and compression

Suppose there's only one fish, and its timer is at 0. Define r(n) to be the number of fish after n days with the above starting conditions (so r(0) = 1, r(1) = 2, ...). Given a closed form for r(n), we have a solution for an arbitrary number of fish with arbitrary timers: just multiply with the number of fish and shift by timer value.

How to find the sequence r(n), for an arbitrary n? There's an advanced open-source CAS (though this one is much more than the others, as it's based on a well-designed general purpose programming language) called Fricas, and it has a package (called Guess) for guessing different kinds of formulas for sequences (or their generating functions). Fricas is able to handle sequences of arbitrary type (e.g., sequences of polynomials), but in this case we have a mere integer sequence. We need a list of the initial members of the sequence (r(n) for n from 0 to 20, for instance), then giving a command like guessPRec(list) results in the following recurrence: f(n + 9) = f(n + 2) + f(n). (Note that it's probably not too difficult to find the recurrence "by hand", but why do that when we have Fricas!)

We could use this recurrence directly for computing the solutions to Day 6, but instead my idea was to get a more direct way of computing r(n). Fortunately, this recurrence is a homogeneous linear recurrence with constant coefficients, and finding a direct formula for it is bog-standard: finding the roots of a certain polynomial equation (called the 'characteristic' equation) and solving a system of linear equations. This could be done in any language (e.g., Julia or C++), but I decided to just stay with Fricas.

After these computations are done, we have a solution for Day 6 whose run-time does not depend on the number of days that we observe the fish! Well, the code I link to below uses arbitrary precision floating point for representing real numbers, which means the dependency might be there for this particular implementation, but the same computation can be done with the "usual" fixed-size floating point types (double in C, for example), too.

Reference for recurrences: https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients

My Git repo, the rest of the solutions are in C++20: https://gitlab.com/nsajko/adventofcode

My Fricas code: [https://gitlab.com/nsajko/adventofcode/-/blob/master/aoc_2021/

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

They’re on standbi

Buenosdillas

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

[Removed]

You take away their little brooms

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

Why

It’s pronounced “Noel.”

After all his first name is No-vac

What, then, is Chinese rap?

Edit:

Notable mentions from the comments:

• Spanish/Swedish/Swiss/Serbian hits

• French/Finnish art

• Country/Canadian rap

• Chinese/Country/Canadian rock

• Turkish/Tunisian/Taiwanese rap

There hasn't been a single post this year!

(Happy 2022 from New Zealand)

Nothing, it just waved

Bob

So that I could frequently say, "I am going to walk 5 miles now."

Edit: My most popular post on Reddit! 😀 Thank you for the awards.

Just to clarify, 12345678

Me grabbing a soda from my (what I thought was) half full 12pk...

Notices there's only 2;

Me: "Awe man... This is a damn bird box!" Her: "What the hell does that mean?!" Me: (Pulls both cans out & shows them to her) "It's only got Toucans."

I'm not ashamed to admit the look on her face was glorious.