How should you arrange 7 water fountains in a mall to minimize the longest possible walk? 

capitalism!!!!

Hey, how to maximize still a mathematical problem

@@lfestevao Easy. Stick them all in the same place, somewhere along the edge of the mall.

@@TacoPotatoYT Make it infinite by just not building them

@@baranxlr But then you have 0 fountains, which is less than 7

I came during the MajorPrep days so it's still weird to see the name "Zach Star" and especially "Zach Star Himself"

bro i thought i recognized his voice but I TOTALLY thought it had to be a different person my god

In Taiwan, everyones an engineer. Its like being a carpenter in Jesus' time. It means a lot and nothing at the same time.

This makes more sense as water fountains theory is to maximize walking

@@duckymomo7935 Why would fountains be about maximizing walking?

@@jursamajso that people walk more through the mall, causing them to buy more

@@kommissarjunior9298 I guess I should have mentioned here as I have in other comments: this problem is not about a *shopping* mall. It's a mall like the National Mall in DC: a kind of park. Buying stuff isn't an issue.

@@duckymomo7935 No. Code requires a minimum; there are not requirements for solicitation.

“The hexagons *are* the bestagons“ -CGP Grey Heres a video to the link: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-thOifuHs6eY.html&feature=share

Me too :)

Yeah, the only thing taking a bit of calculation is how to then place them and what the distance is

hexagons are the bestagons

Hexagons are the bestagons, as they say

I think the reason for this is that we are solving in Euclidean 2-space. The 3-graph would be any easy solution but you can’t represent that in two dimensions, you need curved lines. It’s also generated easily by symmetric group actions, whereas odd-numbered disk covering graphs are not.. idk just some thoughts

Coming back to this i think it might generalize to a čech complex.. so I guess this problem can be generally solved but the computation of a čech complex from the network graph is reportedly “nontrivial”.. I wonder how quickly you can solve the problem for n fountains?

It's not that hard. Intuitively you want 1 in the center and the others arranged in an equidistant pattern. The hardest part is getting the specific distance from the perimeter that the remaining 6 have to be for the distances to be optimized. There are numbers such as 20 or 50 that are less intuitive because you know you have to layer them in rings to optimize the distances but the question is how many rings and how many fountains in each ring.

Intersection points along the radius of the solutions to these cases seem to form perfect star graphs... why couldn’t graph theory be directly applied to this problem? With v = fountains and e = paths.. seems like the problem gets a lot easier.

when you say "best", is that an opinion or mathematical terminology? (like how "regular" is an opinion word but also means something mathematically)

@@mistabean9272 if you mean best as in the tiling that maximizes surface area vs length of the sides, then hexagons are indeed the best tiling of the plane.

I'm sure you're probably aware of these limitations, at least slightly, but its important for everyone to understand gpt's limitations as a tool: Chatgpt will likely give really good and accurate answers to problems that have been studied in depth before, problems with lots of papers written on them and such. This is because its not working from scratch to solve it, its just repeating what other people have written. If you change the problem slightly and subtly it can get confused and just answers what it knows. For example -- I replaced the word 'minimum' (as in minimum distance from water fountains) with maximum and 4.0 didn't pick up on it - it didn't even begin to correct it. In this case, the correct answer should be to place all of the 7 disks on the same point in the perimeter, but it just gave the same answer for the minimum walking distance while using the word maximum. At one point it said to maximise the minimal distance someone could walk which is clearly nonsense. (well, it makes sense, but since the water fountains have to be placed within the mall, the maximum minimum distance anyone will have to walk is 0 for any answer) If you ask it a novel question then it can very quickly get very confused as well, kinda like "there are 100 light switches in front of you. They are all off, and have their own individual switches to turn them on and off. What is the least number of switches you will need to flip in order to turn them all on?" gpt 4 got this correct for me (100 flips, obviously), but 3.5 went off on a prime number spiel. If you word it a little more confusingly (especially if you word it to look more like a coding interview problem) then you can trick 4 as well pretty easily too, as it suddenly picks up on the patterns normally seen in coding interview questions and starts going off of their answers to unrelated problems. The question is very simple, but worded in a slightly more complicated way than you may expect -- you may think "well that's not fair, its a very simple question of course it'll get confused" but part of Chatgpt's usage is to help people learn, or help guide them in unfamiliar areas. This will include a lot of simple and obvious questions that may otherwise be easy to answer, while still being unknown to the learner (by no fault of their own)

​@nicholasdavies5297 quite. It cannot, and does not perform mathematics. It is a language model. You may as well ask Stephen King to teach you math.

​@@nicholasdavies5297 Just wanted to say, this was an amazing read, thank you so much for the effort you obviously put into it! 💙

You mean the web results interpreter?

How about a square?

How about a klein bottle?

Thomson problem for ya!

planet-sized malls

I imagine it wouldn't, as the water fountains could become the perimeter, then just repeat until you reach the middle

Yeah, right about before 1:40 I was like "oh, draw a voronoi diagram to show you which fountains are the nearest fountains in which parts of the mall"

I also thought of them. I suspect that with a basic annealing function you could iteratively find the optimal solution, but the algebraic solution described in the video is also really neat

Who tf said anything about your intuition? I call bullshit

ummm yeah your intuition was right because that is the obvious part????? literally what else can you do besides that.......... the actual "problem" to solve is determining the optimal r ie the distance of the hexagon points away from the center

with water fountains represented by points on a coordinate plane

I GET IT I GET IT

they knew the hard solution

Ah my reply was snapped ;-; did the code, found some estimates for small N (number of discs) (big radius is 1000m) N=1 -> r=1010 (min is 1000) N=2 -> r= 1010 (min is 1000) N=3 -> r=866 (min is 866) N=4 -> r=711 (min is 706) Will post link to code in next comment :) In hindsight I think if each centre was a particle which repelled all other particles (think charges), within a quadratic potential well it could end up settling to these solutions aswell

colab. research. google. com/drive/13ermNlEG4mqnsKCSfm2w_vIZJhgfNGTr?usp=sharing

@@MochiClips Hmm... but do you really want them to repel *all* the other particles? Or just like, the ones where one wouldn’t have to pass closer to another particle before reaching, or something like that?

@drdca yeah all of them otherwise it would just settle with the non repelling ones overlapping :)

@@MochiClips I was thinking which ones repel would change over time depending on the configuration

After the video: Well, I was at least half right... The diameter I guessed, was clearly wrong, but the general orientation of the fountains was correct.

... Can't see how that's related at all

This is a mall like a park, not a shopping mall.

@@jursamaj I've never heard the word used like that. I had to look it up to make sure it was actually a real contextual usage. I've lived in S.Florida, W.Tennessee, and Virginia Beach, and I've *only* ever heard "mall" used in the context of a shopping mall, so this is very interesting. Is it a typical Commonwealth usage? Or maybe used in other regions of the US like further North (eg. New England)?

@@omargoodman2999 The main usage I know of is for the National Mall in DC. Shopping mall is a pretty modern Americanism.

@@jursamaj Facinating. So, I looked up the etymological history of it. It derives from the game Pall-Mall (basically Italian croquet) and the field/alley on which it was played (the "Mall"). That eventually developed into any open field, plaza, or concourse between buildings where people could walk or gather, often "parkified" with grass, trees, decorative sculpture installations, etc. Then from there, a Mall, which was just the open area amidst buildings (eg. government administration buildings, office buildings, etc.) developed into the concept of a Shopping Mall (explicitly a Mall surrounded by various stores and other commercial facilities). This could be free-standing stores, but would more often be a contiguous building owned by a single agent or jointly by a set of agents and shop units could be rented out as real-estate. This could either be an open Mall where the central concourse is open and gives access to shops with inward-facing public entrances, or it could be a closed/contained Mall where the concourse, itself, is sheltered as a complete building with the shops integrated into it. And, of course, in keeping with this etymological history, just as "Park" used to be a grassy area where people could stop their horses/horse-drawn vehicles; but as horses gave way to automobiles these grassy areas became recreational areas for people while retaining the name "park", but the action of stopping your ride (parking) was transferred to cars and the large, open parking lots there for used... we have Malls which used to mean parks, surrounded by Parking Lots which used to mean Parks, and you can get there by driving on a Parkway which used to mean a Park... Isn't language _fun?_

@@omargoodman2999 Indeed, the evolution of language leads to many peculiarities. :)

It ended up being pretty long of a comment...

Guess 3 at the start, than 4, 5, (use the solutions from the circle problem, and try what works best)