How should you arrange 7 water fountains in a mall to minimize the longest possible walk?
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- Опубліковано 13 бер 2023
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In real life, you wouldn't have these problems: The architects were told to _maximize_ the average distance to the nearest water fountain, so more things would tempt people to buy more as they walk there.
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
Holy Shit! He found the password to the channel!
Oh yea, Zach is an engineer, as well as a comedian
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.
sounds like the distribution of cell phones towers for covering a small town.
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.
Intuitively figured a hexagonal setup with a fountain in the middle was optimal. Turned out to be right, neat. 👍 ✌️
“The hexagons *are* the bestagons“
-CGP Grey
Heres a video to the link:
ua-cam.com/video/thOifuHs6eY/v-deo.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
It's funny how intuitively you would think seven circles would have the most difficult solution (since people commonly think seven is the 'most random' number). But it turns out it has the simplest solution of all. Or at least the simplest non-trivial one.
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.
Divide the mall into sevenths, separated from one another, then flood them all with water going in and out for 7 mega fountains that everyone has to wade through to get around.
The cases for 5 and 6 are actually kinda fun except for the end bit, I wish you at least glossed over them more. For 5, we follow the same strategy. We want 5 equally-spaced intersection points along the circumference, so they subtend 72 degrees. We want to find a point which is (1) along a radius that passes exactly between two of these intersection points, so a line connecting the two will be perpendicular to this radius, and (2) for now at least, equidistant from the center and an intersection point. We find that this forms an isosceles triangle 36-36-108 degrees with long side R and the similar sides (root5R -R1)/2, or 0.618...R. This is close to optimal. The idea now is to shrink the circles, maintaining the intersection at the center, until the circles are less overlapping and are tangent, bringing the distance to more like 0.6094. That math is wicked hard, but visually it's easy to see what is done.
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.
At the start of this video, my instinct said a hexagon with a center point too. That is 7 points and i just know of it from a little reading about circle packing a long time ago. But i would have no idea how to prove optimality.
I asked GPT-4, and it knew to put the 7th fountain in the center, but told me to put the other 6 on the perimeter of the circle. It said the minimum distance was 1.15 divided by 2. When I asked if that is true, it apologized and then told me that the fountains go slightly inside the perimeter and the minimized distance is 0.5
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?
What is very interesting in the final solution: It is not only the outer perimeter, that is covered exactly by the outer 6 fountains, it is also the perimeter of the inner fountain, that is exactly covered by those 6 fountains.
If you connect the intersections of the circles, you get perfect hexagons of side length 0.5km and diagonals of 1km. And hexagons (or 6 equilateral triangles) are the best shape to fill a flat plane.
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.
Now do it on a sphere surface!
How about a square?
How about a klein bottle?
Thomson problem for ya!
planet-sized malls
i found this guy with his skits, so him teaching complex math problems hit me like an educational freight train. nice.
Was expecting something involving voronoy diagrams. Could you do a video on them please
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
I realze the mall is just a convenient way to teach the math, but it us fun to hear about and good to know that people still know what stores actually are. Imagine the problem wittern as "How to design an online shopping eebsite to minimizw the furthest dsytance between 7 buttons within a circle"
3:25 "it should be apparent that ..."
It is, but only once you said it :D
As other people have also already said, my intuition was to have one central one, and the others arranged in a hexagon, which turned out to be right.
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
I watched this right after CGP Grey's Hexagon video haha. This is a really high quality video and I enjoy it a lot! Thanks for posting:)
I loved this problem, first we can try to solve by our own and see if we are right, but even more important, this type of problems are really useful for different problem in engineering, like where to place fire exits or roof drains, I hope we can see more of these problems. Thank you.
I could definitely see this problem becoming way harder with like 46 water fountains
I imagine it wouldn't, as the water fountains could become the perimeter, then just repeat until you reach the middle
It was explained a bit at the end, but the motivation for why a circle should be put in the middle was a bit lacking, especially as the minimal example provided to help solve the problem didn’t need a point in the middle. Cool problem, and generally well explained
How not to sound nerdy in the first 10s of a video:
1. Definately don't start with: ' Imagine we have a shopping mall in a shape of a perfect disk ...'
with water fountains represented by points on a coordinate plane
It would be a more practical scenario if the circular area represented a park, since most buildings aren't circular but outdoor park areas can be
love your style of elucidation. care to tackle some fluid dynamics next? or some trebuchet physics?
Great video! I love these puzzles
I love your videos! starting mech eng this winter!
I arrived at the same solution from a different direction, with 6 abutting equilateral triangles surrounding the centre of the mall. The size of the triangles was determined by looking at one of them, and constructing a line from mall centre through the centroid of the triangle and extending beyond the outer side an equal distance from triangle centroid to outer side, to reach the mall boundary. Some simple Pythagorean geometry gave a triangle side length of √3/2km, so the 7 fountains would be arranged with a central one and 6 at a distance of approximately 866m from it and spaced at 60° intervals.
Truly something worth calling a "discovery"
I GET IT
I GET IT
Since it was 7 to fill a circle, I thought of hexagons, since they fit together and are relatively close to forming a circle. The difference between the hexagons and the smallest circles they fit in would be the overlap between circles, so you can ignore that on the inside, it's just covering the whole outer circle I wasn't sure about.
Hey Zach, do you have any book recommendations for an EE student trying to grasp electrical circuit/ circuit analysis? I feel I don't learn anything from my professor as all he does is algebraicly moves variables around for lecture and doesn't explain anything. So I need to learn on my own. Would appreciate any recommendations.
Shoot, just happened to be studying a 2d poisson random variable problem. This helps a lot!
ah yes the disc covering problem because zach just dis covered his yt channel password
Now do the disk covering problem with one disk!
Just when I wanted to comment "prove it" you gave the proof! thanks!
This reminds me of the six-flags booth where you have to cover the full circle with only 5 metal pieces to win a ps5
they knew the hard solution
This feels like it could have an interesting conceptualisation as a convolution over r.
To minimize the longest possible walk between the 7 water fountains, you can arrange them in a hexagonal pattern.
This pattern is known as a honeycomb pattern and is often used in urban planning as it maximizes space utilization and minimizes walking distances.
To arrange the 7 water fountains in a honeycomb pattern, you can start by placing one fountain in the center and then placing the other six fountains around it in a hexagonal shape.
imagine how much planning goes into placing a COVID handwashing sign that begs the user to wash for a very long time but the faucet controls are manual, one-handed, and only runs water for 0.2 sec each press
Thinking about how you could make an approximation using a simulation where the fountains can move around and are repelled by one another.
I don't think this solution is optimal b/c it looks to me like the probability of being close to a fountain is not maximized. There's some areas that would be close to a fountain that are outside the mall... Would be interesting to see an analysis based on this giving weight to the probability and the distance traveled.
Man this is so simple yet still so hard. Surely a predicted population density map as well as a layout of walls would make it more realistic and applicable. And of course as a classic engineer I will solve it via computationally brute force.
Thank god, I was just working on my 1 kilometer disc mall and was wondering where I could place the water fountains.
Now how do I distribute the rest of the stores and elevators?
Tried 7 evenly spaced fountains, got around 0.555km walking distance max. Tried 1 in the center with 6 evenly spread, 0.5km max walking distance (with centers placed at sqrt(3)/2 from the center, for extra fun.) No idea how to prove that that is optimal though.
Would the placement be contingent on how you randomly select points within a circle? There are multiple ways to select "random" points within a circle.
1. Random Angle and random distance from origin
2. Random x,y, resample if (x^2 + y^2) > 1
I think the proposed solution is accurate based on the title of the video, but there might be slightly different solutions based on the minimum average distance.
... Can't see how that's related at all
I wonder how this can relate to the traveling salesman problem
After taking geometry, I love how I finally understand what you’re saying.
I didn't solve it mathematically, but still got a similar solution, a star made of two triangles but touching the circumference would be dumb so at around 75% closer to circumference from centre and obv 1 in centre
My initial estimate is one in the centre and six sixty degrees apart on a circle with a radius of ~700 metres. Now I am curious how far from the optimum solution that wild guess is...
Update: Turns out my general idea was OK, but the distance of the perimeter fountains from the centre was bigger.
And then, draw lines from each fountain to the closest fountains, and at the midpoints of those lines put drink vending machines or a food court. Also, put the entrances to the mall at the points where the disks intersect the mall perimeter and put vending machines there, too.
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. :)
Zach I would love to have the Desmos link that you used to create the circles.
I feel some code could potentially converge on a local maxima (though proving that is a global maxima is a whole different question lol). Kind of stochastic gradient descent in the (2N+1)-dim space.
Only an approximate solution but basically set up nodes at fixed distance intervals eg (x,y) where x and y were integers within a 1km radius of the centre (for easy counting and area approximation).
Suppose there are N discs each with centre position (x_i,y_i), and all have radius r. Then
Pick a random initialised starting position and define the cost function
C(X,r; L) = -(Total area covered) + L r^2
Where L is some number (which say starts at 0, but is gradually incremented)
Where (Total area covered) is just the proportion of the nodes contained in some disc
Then either
1) update all the positions by adding a random small Normal variable to each, along with a random small normal variable to the radius (maybe not needed). (larger variance, should 'move' faster but may not converge very quickly (on each step will be of size "epsilon^2 chi_squared on (2N+1) degrees of freedom") OR
2) Pick a random value to update, and add the small normal variable. (should be slower to converge but give a more stable answer maybe?) (each step will be "episolon^2 chi_squared on (1) degree of freedom" so much smaller steps in the (2N+1)-dim space with Euclidian metric) OR
3) Some combination of 1 and 2 lol
Sample say 100 of these steps new positions and pick the one that minimises the cost, rinse and repeat.
Once say the whole disc is covered increase the L value so that it then focusses on minimising the radius (I think this could be nice and visual)
Observe what happens - the way I've suggested setting it up would initially focus on covering the whole sphere, likely increasing the radius until whole thing is covered, then as L increases will try more and more desperately to nudge to minimise the radius needed.
Might knock the code together but that should do the job
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
Cool application. Things like this could also be applied to fire hose or extinguisher locations.
Your math videos are really cool! I'm always waiting to know more about interesting things, and the way you present everything makes the experience even better 👍I've kind of been wondering, in some of your vids you talk about trig an such. But one thing I don't quite get is how exactly sin and cosine translate to rotation (ex: the function rotation substitution: x′=xcosθ−ysinθ y′=xsinθ+ycosθ, theta being amount to rotate in a clockwise direction). Do you think you could make a video on this? It's probably presumptuous since it probably takes ages to research and edit... I would really appreciate it though! UA-cam doesn't really answer the question well after all... can you imagine the applications?!
It ended up being pretty long of a comment...
how would you solve a case of MORE than 7, where the radius is so small that the inner circle is not sufficient to cover the ring of outer circles? How do you find the precise position and number of "inner" circle which do not touch the largest outer radius to sufficiently cover 100% of the area?
Guess 3 at the start, than 4, 5, (use the solutions from the circle problem, and try what works best)
Nothing like a little Zach Star to start off my day, I think I speak for all of us!
"Ok, imagine we have a shopping mall built in the shape of a perfect disc..."
Weird ass skit opening, but you just know the punchline will be all worth it!
The shortest distance between an infinite amount of points is the circumference of a circle. If you take any amount of randomly distributed points and try to adjust their position around the center of a circle, you'll find the shortest distance in the polygon's sides.
I can't unhear his skits
Using just my intuition if you generalize this problem to any n circles covering an area of minimum distance of size k, this problem is actually NP-complete and can be reduced to NP-hard Vertex Cover. Kinda interesting
I instinctively knew this was going to be the maximal solution. Hexagons are the bestagons.
Engineers solition: draw a Circle with a bit over half the radius and space out 6 fountains evenly. Put the last one in the center. Distance is about r/2 and a bit
Without watching the video, here's my answer. Place the water fountains along an imaginary circle 0.5km radius sich that the fountains form the vertices of a regular heptagon.😊
Hexagons are the bestagons!
My first thought before watching the solution or calculating anything:
One fountain in the center and the other 6 fountains in a circle with equal distance between them or a hexagon with the fountains at the "tips" of it.
The circle/hexagon should be centered at the center of the 1km radius circle and should have a diameter of 1km (radius = 500m).
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.
I clicked on this with the understanding of them being water fountains that like shoot water in the air and you throw coins in to for a wish. Was very interesting to find out I was wrong 2-3 minutes in
2:38 and 4:45 both remind me of the roots of unity, the 2nd one much moreso.
Wonder if that'll be relevant if I keep watching more lmfao
1:52 "The maximum that someone would have to walk is 1.12km." That is a big mall 😮. Idk if malls are actually that big, I don't go shopping, but it seems like going to such a mall would be a workout in and of itself 😐.
What I first thought was to create a regular poligon with the number of sides being the number of fountains and then going on the midpoint of every one of the sides and putting a fountain there. Could someone please explain to me why this work or why this doesn't work?
my professor did a research about this and published recently! It was about the shortest distance to a point in an sphere to any n amount of points!
Martin Gardner wrote an excellent article on the 5 disc solution.
His explaining voice in this sounds like his ranting voice in the comedy videos
I would create distance from center half of radius. angled at 360* / number_of_fountains. Simple.
In this specific example, I would have the first fountain 0.5km from center at true north, then id have another fountain 0.5km from center at 51.23* east, and so on.
The fact that the optimal solution was the instinctive solution for me
here, you find the optimal solution for the maximum distance traveled to the nearest fountain, but does the solution change for the minimum average distance along all points in the circle?
Back on the scene !
and as always, Hexagons are the bestagons!
The intuitive solution is to use hexagon, the bestagon. As always.
Now on to a mall with an irregular shape and lots of obstacles. As well as 2 floors.
And some water fountains that are not built for many people to use at the same time, having longer ques, making them less optimal.
So let us represent them by time it takes to reach the fountain, to include wait times in line.
Small water fountain = 3x time to reach.
When Zach realises that he has another channel to post stuff
2:35 some things are more risky than others, for instance a submarine ride to the Titanic is more risky than the typical commute to work.
My solution? This looks perfect for some gradient descent.
You will get stuck on a local minimum. Finding a global minimum with a large number of disks is very hard and computationally intensive
Ah yes, when you don't know how to solve a problem just apply the modern solution of slapping a 2 trillion parameters neural network to it and hope for the best.
I'm 30 seconds in, so here's my train of thought:
-- 7? That's seems oddly specific.
-- I know Hexagons have a special quality (being made of equilateral triangles) so maybe the six points of a hexagon with the seventh point in the centre of the circle. Yes, intuitively it makes sense to have one of the fountains directly in the centre.
-- There'd be no point putting the points of the hexagon on the edge of the circle, might as well bring them in since they're serving more area than being on the edge.
-- Thing is, if we bring the points in halfway from the edge, someone at the edge of the circle would need to walk half the circle's radius, whereas if they're between the point and the centre point, they'd only need to walk a maximum of 1/4 radius.
-- If we place the hexagon points 1/3 of the radius in from the edge of the circle, then the longest distance anybody would need to walk is 1/3 the radius.
-- I'm not sure how that would work for people equidistant between two of the hexagon points, but it's probably not far off 1/3 radius.
Now I'm gonna watch the video to see how I did.
EDIT:
Okay I was close. I had a feeling the gaps betwee nthe points would cause it to be r/3, and I had a suspicion it would be r/2, but I didn't find the reasoning for r/2.
I wonder about the case where the number of points is 2. If you place them both not in the center of the circle, then there will be a scenario where you have to walk more than r.
Theta_max = pi => r = 1
So having one fountain is just as "bad" as having two when the only qualifier is this minimization
What a pleasant mall!
We could make these fountains to walk and the problem doesn't exist.
this is really cool
I really thought this would end up a golden spiral
I came up with separating them equally on 7 symmetrical rays such that the max distance to the top of the ray is equal to how far out on the ray they are placed. That gives from the law of sins:
sin(2*pi/17)/c = sin(13*pi/17) / 1
solving for c
c = sin(2*pi/17)/sin(13*pi/17)
or 0.53620899822
This method would be easily scalable to N fountains -- the issue is the damn center one -- that makes it more complicated for sure. You would need to do your approach for that then find out which solution returns the lower result.
Looks like 0.5 is better when one is in the center!
By travelling as close to the speed of light as possible
id assume that we want to have the least number of overlaps of the imaginary circles
I have a puzzle for you. Say your mall is a Golden Rectangle. You are given 8 fountains; how do you space them such that all 8 cover the area with both circles whose centers are those fountains, and regular hexagons whose centers are those fountains, with the pattern between the distances of each fountain being 1 part, 1 part, 1 part, 2 parts, 2 parts, 1 part, 1 part, and 1 part respectively, and minimize the difference between the hexagon area overflow and the circle area overflow?
Put them all at the entrances. Walk in. Take a sip. Return to your car. Leave. This is the proper way to navigate every shopping mall.
6:57 Hexagons are Bestagons!
I use a transformed R2 sequence for realtime applications
Hexagons are bestagons!
so basically, how to use 7 circles(as small as possible) to cover the entire big circle. seems like those super hard questions that have difficult solutions
8:55 Hey, that looks like Kurzgesat!
Alright. I'm rarely one to solve these things but this time I was actually right, wohoo! Of course I didnt do any math but my first idea was "Probably put one in the middle and then the other 6 evenly spaced arround so you cover the people far out". Then I decided that it cant be that simple and watched the video.
How hard would the problem become if instead one wanted to minimize the average distance to the closest water fountain throughout the mall?
This is an equally good question, in my opinion.
Dunno why, but instinctively I think about filling cirle with shapes like pentagram, honeycomb etc.