The Brick Factory Problem - Numberphile
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- čas přidán 30. 05. 2023
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To me, the obvious solution is to fix the track crossings rather than minimise the amount you use them. But maybe that's why I'm an engineer rather than a mathematician...
Yes that was my intuitive approach also.
Or use both techniques to minimise cost/effort/power/etc
How hard you try to avoid crossings surely depends on how much more expensive the improved crossings are compared to the extra track needed to avoid the crossing.
Or make each stopping point a through-station instead of a terminal.
I think fancy crossings would be expensive in build time and capital, longer routes expensive in efficiency and (therefore) cash flow. Case by case decision I guess.
The story of working mathematically in such adverse conditions is inspiring.
Gets you through it.
@@Sonny_McMacsson Exactly, anything to distract your mind from the horrors all around you will only help to endure them. Still inspiring and admirable, of course.
Mathematically minded individuals will spend their time musing over math problems even if they are in a garden of eden.
He was probably mentally grumbling about the stupidity of the idiots who built the place...
he was a collaborator to the system that oppressed him, by doing actions that benefit a system you are helping to support that system. the bricks get used to make bunkers, the more bricks the more bunkers, the more bunkers the more attackers it will take to over come them, by improving efficiency he likely caused greater death of those that would free him. He deprived those that were knocking over the cart and slowing down the system, the chance to fight those that in enslaved them. is it all about you and what gets you threw it, about the entertainment of your own mind, what helps you pay the bills, get a head over the next slave. NEET, lying flat, atlas shrugged, quiet quitting, the cheap slave with to many whip scares, the lazy. these are the actions of those that sacrifice their own benefit to hold to their own morals and beliefs.
i think it is outrageous that they didnt use x(n) for the minimum number of crosses. Absolute tragedy!
viewer who crosses their z's: *sweats nervously*
But all these mathematicians draw their 'x's like ")(", so there's no cross.
@@topherthe11th23 I guess people call it a cross 'cause one of the lines crosses over the other, and/or 'cause it's a cross rotated a little ways.
Yeah, this is, like, the one time when x would be the best.
@@wbfaulk rolles theorem: what doesn't cross?
Three kilns and three storage units is literally the three utilities problem!
That was my thought as well!
I couldn't help reciting the dedication of a graph theory book I had years ago: "To Kazimierz Kuratowski who gave K5 and K3,3 ..."
@@pierreabbat6157 Yeah, without him we still would have no graphs anything comparable to those. What an achievement indeed!
Yes, but it is asking for minimizing the number of crossings and not only proving that the number is not zero.
“Just put it on a bagel!”
For those who don't know: these graphs that are split in two parts with mutual edges between their vertices but not within same part are called "bipartite graphs".
Complete bipartite graph reminds me of videos on planar graph
Also for those who don't know: the graphs in my phone gallery are called photographs, and they are used to depict naked images of your mother.
Pretty cool! Graph Theory is fascinating.
@@aceman0000099 lol
@@hafizajiaziz8773 yea it's all graph theory.
so long since I last watched Numberphile… James was my favourite guest every time!!!! Love that he ages beautifully
Glad you're back
To anyone who thinks this kind of maths is a bit abstract - I used to use a lot of these graph algorithms in the EDA (electronic design automation) industry - when you get down to really low level problems, this sort of stuff is invaluable and using it is the only way to make many things realistic and/or feasible.
What firm are you working in? (Working in EDA too)
@@anujthakur614 I worked for a startup called Arithmatica - later acquired by one of the big boys after I'd left. We were developing synthesis tools that specialised in producing low area/high speed gate level descriptions.
Its just graph theory
The obvious solution is to place each brick onto a blockchain and 3D print it on location.
And also throw AI at it
The obvious solution is to place the kilns on the trolleys and rolling them to where they are needed, no need for storage units
If a track merger was a viable intersection for this problem, I think you could bring it down a lot by having them all merge down to one and then split back up.
Pretty sure those were rudimentary tracks, hence the original problem, a track merger should be even more difficult to get right compared to a (somewhat) orthogonal crossing.
That would be a modified Steiner tree problem
I would put kilns on one side and storage units on the outside of a loop, with each kiln and storage unit having an entry/exit ramp.
The number of junctions is just kilns+storage units.
Ooh, I love that application of minimizing the number of layers on computer chips! It's really awesome how these problems that seem like idle curiosities eventually find unexpected real-world relevance.
Am I the only one who feels that the story of the proofs will even be MORE interesting? Why 6? Why 12? What was the mistake that no one noticed for more than 10 years?
agreed, I want to see the story of the proof now!!
The reason that we know that the conjecture for complete graphs is correct up to 12 is because it has been checked computationally. The reason we don't know any more is a because it would take too long to calculate.
Finding the actual minimum is pretty interesting, because it takes so long to check every possibility it is instead rewritten as an integer linear optimisation problem solved using the associated optimisation algorithms.
+
Can't believe I've been listening to James explain maths curiosities for well over a decade now (since high school until PhD)! And this man seemingly doesn't age. Legend!
Why is it dotted?
The line stands apart,
a whimsical stroke,
a work of art.
Why does my heart
dance for that line,
so delicately dotted,
so mystique and fine.
The premise for this one reminds me of Futurama, when Hermes was in a prison camp and optimazed the labor so that it could all be done by one Australian man.
"Quiet mate! Hauling the empty carts is the closest thing we get to sleep."
Put all kilns/storage in a circle. Tracks as spokes to the center. Stop all tracks before they overlap. 0 overlaps, but you have a nightmare region in the center where you have to wheel your barrow without a track. Assign a number of forced laborers to the center region to assist.
I don't know why, but as soon as I saw thumbnail and title I felt this video features Dr Grime
From the sounds of it, a point of singularity are considered as many points has have already passed through it instead of "one".
I'd be interested to hear exactly where some of these proofs that were later proven wrong were discovered to have issues
+
This is just like that famous puzzle of connecting three houses to the three utilities. In fact, for the 3K3S example, it's quite literally the same!
And it can be solved the same way: just add a third dimension, by making the railroads elevate above or tunnel under each other, you can make it have no crossings.
Alternatively, stepping outside the realm of abstract maths, you can connect a kiln to just one storage, and then connect the storages so they can exchange stock when needed.
Or fix the damn crossings.
In practical terms there would be a no crossing, just need to converge all tracks to a rotary platform.
mathematician: "Need to minimize crossing." engineer: "Need to fix crossing."
Only true mathematicians are crazy enough to think of maths problems while working in life threatening situation .
@@Holofractalius Totally agree .Thanks for suggesting.
If you want the most efficient way to do a task, ask a lazy person to do it.
@@Holofractalius Yep, if the lazu person is too lazu to use their brain, nothing will happen. You need to force them to do it!
@@Holofractalius My 6 year old grandson is very bright. So bright he'd rather use your brain to solve problems, not his own. I'm wondering if he has a future in AI now...
Hungary sided with nazis in ww2. He was manager of some sorts, I suppose
Well I timed that refresh perfectly...
Hi Squonk 😊
same!
first comment?!
Wow.
what refresh?
I was fixated on that spiffy travel schedule display on the wall the whole time.
He looked so happy when he got to name the variables after (k)ilns and (s)torage units
10:28 this crossing, if we draw the full graph on a donut, is the crossing passing through the hole of the donut we need to make a map that needs at least 5 colours to colour every adjacent country with a different colour.
This reminds me of the Utilities Mug on Maths Gear.
Another fantastic Numberphile video. I am so glad I found this channel 6 years ago.
What I would do is make it so instead of multiple tracks crossing, I would have the spot where the crossings would happen cut out and replaced with a single track on a turntable so that you just orientate the track on the turntable to align with the track you need to use.
I knew K_3,3 and K_5 were nonplanar, but it had never occurred to me to think how many crossings were required to draw them on a plane. Has this problem been studied on any other surfaces? I know K_3,3 and K_5 can be embedded on a torus (and hence any surface of higher genus), and as for nonorientable surfaces, the Möbius band (and the Klein bottle) and the projective plane.
Just a thought: Despite not having the crossing number cr(G) for a given graph G, an upperbound N will tell you that G can be embedded in a surface of genus N. Since such a surface will have N "handles" which can serve as bridges when you embed the graph G, which allows you to avoid crossings. So to me it seems the problem of crossing number is equivalent to finding "the best surface" it can be embedded in.
Robertson and Seymour actually have one of the best theorems of graph theory (I think): For any surface T, there is a finite collection of graphs H such that a graph G is T-embeddable if and only if it has no h-minor for any h in H. For example, on the plane (S^2), H=K_3,3 and K_5.
Following this theorem, if you look at minimal surfaces a graph is embeddable on (characterized by the number of holes in the minimal surface), this problems and the one in the video are equivalent.
@@beardedboulderer2609 Thanks for the heads up! I will absolutely check out that theorem.
There is something called the "toroidal crossing number," which is what it sounds like. If you look at papers on that subject, you'll also find some work that has been done for other surfaces.
@@beardedboulderer2609 came here to mention the Robertson-Seymour excluded minor theorem! Nice work.
So computers chips are full of kilns, bricks, storage units and railway tracks? I've always suspected as much.
Love it when Dr Grime talks about classic problems and graphs! Bonus point if it's not proven yet.
I like to watch the post-credit (post-sponsor?) scenes showing outtakes or bloopers. Too bad this one has none, haha...
sounds like the book-page graph problem is a special case of the everything to everything version of this problem
2 simpler solutions:
Arrange all the kilns / storage units in a circle (ideally alternating, but whatever). Then run all the lines to the center point of the circle, and install a turntable / circle junction. Then any kiln can connect to any storage unit with 1 single crossing.
Or, do something similar, set up kilns / storage in a circle and radially extend from the center. Set a loop around the outer perimeter of all the kilns / storage units. Then each spur connects to the circle in 1 place, so there would be (in your 6x5 example) 11 crossing points, which is far less than the 24 you suggest.
The engineer in me can't help but think "If intersections are so bad, why don't you just design the rail system with switched junctions instead?" You could have all the kiln routs converge to a single line that then diverges to the storage units, the key difference is that the forks in the line would be controlled by switches that would make it so the carts are only ever contacting one set of rails a time. Perhaps this would reduce or eliminate the instability caused by the intersections. Another possible solution: use rubber tired carts not rail carts. Well anyway, the point is to make a mathematical puzzle, and even if that problem isn't actually practical to the brick carts, it can apply to other situations like the circuitry problem.
I noticed that the 3 kiln to 3 storage unit case of the problem is identical to the 3 houses and 3 utilities unsolvable puzzle. I remember as a kid my uncle in Russia posed to me that puzzle, and very quickly I conjectured it unsolvable (the goal being to have 0 intersections), at least on the Euclidean plane, but I really wanted to mathematically PROVE it was impossible and felt frustrated that I didn't have the mathematical tools to do so.
My first thought is to connect to a turntable, next would be switches.
My first thought is "don't ask a mathematician to do engineering"
Maybe that was the plan, and lots of bricks are needed to buy that upgrade.
As i said to the other commenter with the same idea, having tracks converge is actually more difficult to do compared to a somewhat orthogonal track crossing. So if the cart tracks at the work site were so rudimentary, that they could not get the crossings right, they for sure could not get tracks to converge. Remember, this is not a rail way for freight trains, this is a work site, possibly even just temporary, and since it's forced labor, they won't really mind some worker there having to pick up a cart and pick up bricks, it's certainly cheaper than having to install proper tracks.
@@yt.personal.identificationSecond thought: don't ask an engineer to do mathematics.
So there's a (proposed) formula for doing it where all type A must connect to type B, and a general one where all must connect to all, but is there a general one where all type A must connect to type C, all C to type D, etc.?
At 2:37, the count of crossings is reported as 7, but typically I think of such problems as disallowing three edges crossing at a single point - is that actually part of the problem, or just a simplification?
That must be disallowed, otherwise all graphs could be solved with only one or two crossings. Just send everything into the rail vortex.
Always a pleasure to see Dr. Grime!
James Grime with the first pose full of style sheesh
Imagine he comes back to his boss like: i solved it but it only works in 27 dimensions
It had to be a mathematician that when put in a forced labour camp thought "how can I optimise productivity for my captors?"
It's nice that we humans can think of things to distract ourselves from horrible things going on externally. There are many ways to fix the brick foundry
Missed opportunity to bring the complete graph segment back into the kilns and storage segment. The kilns and storage part is trying to make a complete bipartite graph with minimum crossings.
A bipartite graph is a graph where there are 2 sets of vertices (e.g. kilns and storage units) where members of either set only connect to members of the other set, for those wondering.
Oh snap, Singing Banana is still at it, all these years later! I'm so happy about that :)
Just connect all the storage units together and run each kiln to one storage unit in most cases then they can shuffle between units.
This is kinda trivial but you can always bound the number from above by creating a roundabout, but only if you allow that type of intersection.
Now im interested in how they calculate the minimum amount of layers for computer chips :P
You work in 3D instead of 2D
I think he means PCB's which can be multi-layered.
(what is true is that some ASICs technologies allow for individual layers to be rotated against each other in non-orthogonal albeit fixed angles, especially very high density flash memory & I've seen one which uses an interconnect layer that was yawed by (1,2) grid units (aka knight's move rotation) & another one by (1,3) grid units.)
@@FrankHarwald oh yea i didn't even consider how tough it is to plan them if the orientation of the circuit is so restricted :0
It's incredible how I could listen to James "Weasley" Grime all day long
Excellent explanation thank you so much.
You’re kiln-ing me with these extremely fascinating videos!
Would love to see a sequel to this that includes more dimensions, or plots the scenario on the surface on a sphere or manifold.
I’d love to hear about more real world applications for this sort of thing! That circuit example was brilliant 😊
The brick factory problem happens when I don't eat enough fibre
Or drink enough water? Have you tried magnesium supplements?!
Another obvious solution is to utilize the fact we're working in three dimensions, and have a bridge over the track.
The reality of moving brick is less complicated when you have more than a two dimensional space to run track within, such as running some track under/above other tracks, and the starting places are more than single points, but areas. So your proofs are extremely useful for getting best efficiency for layers.
True, PCB design is as much art as science and the more layers you have, the more expensive to manufacture.
If my choices are between maybe derailing a cart when I cross, or being _guaranteed_ to push a cart (full of bricks!) up a steep hill every time I cross, I'd probably take the chance of derailment.
@@fieldrequired283 Yeah, but we don't worry about that in this century... we push the button or flip the lever and the Minecraft minecart goes ZOOM!
Also the reality is more simple when you realize you don't need everyone to connect to every other. Each connecting to two or three gives you lots of flexibility for load balancing. It probably gets more complicated when you realize you need tracks from every storage to the loading area for them to get picked up by a train.
When you suggested a two dimensional space to run track, my brain immediately jumped into homotopy theory/∞-category bs with "but now you have tracks between the tracks".
This would be like having surfaces which you could slide your tracks along to reposition them (keeping the endpoints fixed), and now you would be interested in minimizing how the surfaces intersect.
James has the most clickbaity problems. He seems like the most interesting mathematician. Can't believe i've watched him for over 10 years.
"In the early post-war years, the streets were patrolled by soldiers. On occasion, random people were seized and sent to penal camps in Siberia. Once such a patrol stopped Turan, who was on his way home from university. The soldiers questioned the mathematician and then forced him to show them the contents of his briefcase. Seeing a reprint of an article from a pre-War Soviet magazine among the papers, the soldiers immediately let the mathematician go. The only thing Turán said about that day in his correspondence with Erdös was that he had "come across an extremely interesting way of applying number theory...""
With the stuff I'm working on I'm still frustrated graph theory hasn't been fully explained yet.
If you connect all of the Kilns and Storage Units to a large Round About (does not have to be circular or symmetrical), then there are no track crossings. However, this introduces other issues, such as the travel distance will be greater between many locations.
Just connect all points in a circle and have the train drive around in circles and make it stof wherever it needs to load/unload
A new numberphile video with James Grime? Obviously a no-brainer to watch it immediately!
The realistic solution is to just reuse the tracks. You can just make a roundabout or go through another point.
Yep. Use a combination of wyes and through stations. Loading and unloading is done on a siding, not the main line. Even including a freight station for raw material input and finished product output, you can solve this with no diamond crossings at all.
It's not really the spirit of the mathematical question though.
Mathematician: Let's redesign the factory so we don't spill bricks.
Engineer: Let's redesign the crossings so they don't cause brick spills.
Me: Maybe just be really careful?
Dude was forced into slave labor and still managed to turn it into a mathematical exercise, absolute legend
Using your brain like that helps keep you sane. There are PLENTY of stories from POWs about such distractions.
3 kilns and 3 storage units is a tradition around here. Grant Sanderson enters the the video with his coffee mug!
You have only so much space for a factory. Paving everything with rails is a bad idea.
1. More wheels under the trolleys will make them more stable. Redesigning the track may also help.
2. Dividing the factory into several parts, with their own furnaces and storage (2x2). Making them back to back in an alternating pattern (checkerboard) will add more flexibility, but won't create any crossings. No chaotic tracks. Easy to expand in the future if needed.
3. Discarding the rails altogether.
4. Making storage units bigger to limit their number.
edit:
A checkerboard pattern makes one kiln for 4 surrounding storage units, and vice versa. IMO more than enough flexibility to cover all the needs.
I know the main point is the graph theory problem but the engineer in me is screaming the whole time about the 2-crossing solution for arbitrarily many kilns/storages by merging and then unmerging tracks
I’ll always watch and Like JAMES GRIMY videos
The Brick Factory Problem" presented by Numberphile is a captivating exploration of a mathematical puzzle that might seem simple at first but quickly reveals its complexity. This video does an excellent job breaking down the problem and providing insights into the underlying mathematics. It's a testament to the beauty of mathematics - how a seemingly straightforward question can lead to such intriguing results and open doors to deeper mathematical thinking. Numberphile consistently delivers top-notch content that makes math accessible and exciting for everyone.
I wounder why they did not use a relay point. Basically a storage in the middle where all kilns connect to. A set of workers who were tasked with unloading stuff that arrives from the kilns and distributing it into the storage units. or...just have a bigger storage unit.
Realistically, James, why can't each kiln have its own storage unit, or several kilns share a single, massive storeroom? This would involve no crossings at all. I guess they can, so the problem, although phrased as reality, may have been purely theoretical.
The optimum solution to the 3+3 case is to build your brick factory on the surface of a coffee cup.
Maybe that's where the expression 'bricking it' came from 😂
The description wasn't very clear about whether you're trying to minimise the number of _track crossings_ (i.e., the number of places where tracks cross each other) or the number of times _carts_ have to pass over a crossing.
If you have all points surrounding a twist and pivot crossing you can get away with just one. It will be a high traffic crossing.
We should use the z axis, use a little tunnel or bridge over the crossing points!
Is there a 3d (not like the way you showed the computer chips, as those are still 2d in nature) extension of this? Am I correct that there are always non crossing solutions?
You know, it's kinda nice to see floor division outside of programming.
Why couldn’t you arrange them in a large circle, make a smaller circle of track inside that circle, and then have every kiln/storage unit attach to the circle? Then the number of crossings is exactly equal to the number of kilns and storage units, and you can get to every point on the graph.
I guess we can say, going "through" another kiln-storage is not allowed, you can only have point intersections with other paths.
I think you can bring the number always down to 1 (counted naively) by doing that. A column of kiln and a column of storage with all of their connections routed through a point in the middle.
It's implied that only two tracks can ever intersect at a given point. Or, equivalently, you're counting the number of times tracks cross, not the number of places where there are 1 or more crossings.
@@QuantumHistorian I’m not understanding why this applies. If you have a circle with spokes coming off to each destination, you still only have a series of single points of intersection. They’re not all intersecting at the same point - they’re all converging to the same circle. Like a roundabout.
@@titleloanman Think about it this way, do the paths between different kiln/storage pairings have to share a portion of track/circle? If so, that is not allowed.
@@titleloanman I think to see QuantumHistorian's point, draw the various travelling paths from kiln to storage in different colors and see that the colors intersect many times. By his implied rule those would add to the intersection number.
Though these implied rules do feel a little adhoc from the viewpoint of the original problem and also because these rules are only telling you after you've finished your drawing. In order check pairwise intersection completely you kind of have to look "globally".
If you have more than six kilns and more than six storage units, you’re probably firing the kilns continuously enough to designate half the storage units (rounded up) for unfired bricks and the other half (rounded down) for fired ones.
Haha, but the problem is if the 6 kilns are all specialised to one type of brick, and all the storage units needed an equal mix of all 6 brick types. I don't know who would ever do this insanity though.
Increase the delivery times by the amount of factories and have all the storage in one location. One track, two engine cars facing either direction.
Adding the 3rd dimension and its z-axis should help, right?
Not having them in a plane is a really interesting and probably very frustrating option. Like have middle crossing be on two levels.
Looks a lot like combinatorics, a field I've always found fascinating! Mostly as a poker player...
I love this channel!
After Pal Turan was finished, all of the forced labour in that camp was done by a single Australian man (Futurama reference)
For tracks a turntable would have been handy.
Wow, I was just reading about this problem yesterday!
There is a mistake in the problem description or a better solution is possible. At 2:39 we see that multiple crossings over the same point are only counted once. At 3:17 we see that curved tracks are possible. Given these two precedents, the partial "optimal" solution at 5:45 can be beaten by (for example) getting rid of intersection #7 by running the 4-5-6 track through intersection #2 instead of creating a new intersection.
I've just looked at the original problem. Pál Turán would have counted eight crossings at 2:39. This is a happy mistake because it gives rise to other rich problems. For example - for a bipartite graph with straight edges what are the minimal number of "crossing-vertices" created? At 2:39 we count seven of these "crossing-vertices."
I would desing the factory such that one kiln would be connected to one storage unit, and then the storage uniits would have separate tracks to connect between other storage units. I dont think you need so many tracks in brick factory :D
Everything changes when we start thinking in 3 dimensions for this problem
I appreciate the mathematic angle. Though, the first thing I thought of to solve an increasing number of kilns/storage units was either using the locations as stations which could be passed through, or to add a main line where most travel would take place. At that point, however, it'd be an issue of throughput instead of an increased chance of mishaps(Accounting for the possibility of operator error, as well).
Lovely video, regardless.
taking notes for my next Factorio run
For anyone interested this is related to further maths a level graph theory
I love how they just casually have their channel's stats on the wall behind them
Great problem, great background story
Over all the years, this is the first time I've paid attention to your clothes. Nice shirt.
Hi idk if you’ll see this but thank you for your videos and keep up the great work
The obvious solution is to have storage units linked by a single track so you deliver to the first unit and if required the cargo is transferred to another unit. Same deal with kilns.
There's only a couple of tracks that don't need to cross
Then the mathematicians would have nothing to eat!
The minimum crossing for bricks is 0 for whatever amount you have because you can run the rails directly through the warehouses....just run the wagons in one direction, pick up at the foundry, drop off at the yard, done.