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...

@MarkAhlquist

11 ай бұрын

Yes that was my intuitive approach also.

@mirr0rd

11 ай бұрын

Or use both techniques to minimise cost/effort/power/etc

@ragnkja

11 ай бұрын

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.

@pleappleappleap

11 ай бұрын

Or make each stopping point a through-station instead of a terminal.

@Fidtz

11 ай бұрын

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.

@Sonny_McMacsson

11 ай бұрын

Gets you through it.

@DaTux91

11 ай бұрын

​@@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.

@ddognine

11 ай бұрын

Mathematically minded individuals will spend their time musing over math problems even if they are in a garden of eden.

@derrickthewhite1

11 ай бұрын

He was probably mentally grumbling about the stupidity of the idiots who built the place...

@Amipotsophspond

11 ай бұрын

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!

@christianwolirdeng4766

11 ай бұрын

viewer who crosses their z's: *sweats nervously*

@wbfaulk

11 ай бұрын

But all these mathematicians draw their 'x's like ")(", so there's no cross.

@5ucur

11 ай бұрын

@@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.

@vigilantcosmicpenguin8721

11 ай бұрын

Yeah, this is, like, the one time when x would be the best.

@DrKaii

11 ай бұрын

​@@wbfaulk rolles theorem: what doesn't cross?

so long since I last watched Numberphile… James was my favourite guest every time!!!! Love that he ages beautifully

@numberphile

8 ай бұрын

Glad you're back

Three kilns and three storage units is literally the three utilities problem!

@LordDedenova

11 ай бұрын

That was my thought as well!

@pierreabbat6157

11 ай бұрын

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 ..."

@HoSza1

11 ай бұрын

​@@pierreabbat6157 Yeah, without him we still would have no graphs anything comparable to those. What an achievement indeed!

@JohnSmith-zq9mo

11 ай бұрын

Yes, but it is asking for minimizing the number of crossings and not only proving that the number is not zero.

@tommyhetrick

11 ай бұрын

“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".

@hafizajiaziz8773

11 ай бұрын

Complete bipartite graph reminds me of videos on planar graph

@aceman0000099

11 ай бұрын

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.

@betopostagem125

11 ай бұрын

Pretty cool! Graph Theory is fascinating.

@LoveDoveDarling

11 ай бұрын

@@aceman0000099 lol

@d1ma894

11 ай бұрын

@@hafizajiaziz8773 yea it's all graph theory.

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.

@anujthakur614

6 ай бұрын

What firm are you working in? (Working in EDA too)

@gregreynolds5686

6 ай бұрын

@@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.

@ansumanc

3 ай бұрын

Its just graph theory

The obvious solution is to place each brick onto a blockchain and 3D print it on location.

@jan-lukas

11 ай бұрын

And also throw AI at it

@doncarlodivargas5497

11 ай бұрын

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.

@Jonathanizer

11 ай бұрын

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.

@Mathghamhan

11 ай бұрын

That would be a modified Steiner tree problem

@tinaus646

11 ай бұрын

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.

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?

@joleneonyoutube

11 ай бұрын

agreed, I want to see the story of the proof now!!

@jaredcramsie182

10 ай бұрын

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.

@GerSHAK

6 ай бұрын

+

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.

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!

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.

@kmbbmj5857

11 ай бұрын

My first thought is to connect to a turntable, next would be switches.

@yt.personal.identification

11 ай бұрын

My first thought is "don't ask a mathematician to do engineering"

@Palparepa

11 ай бұрын

Maybe that was the plan, and lots of bricks are needed to buy that upgrade.

@Jonathanizer

11 ай бұрын

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.

@RolandHutchinson

11 ай бұрын

​@@yt.personal.identificationSecond thought: don't ask an engineer to do mathematics.

mathematician: "Need to minimize crossing." engineer: "Need to fix crossing."

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.

@brianrose85

11 ай бұрын

"Quiet mate! Hauling the empty carts is the closest thing we get to sleep."

I don't know why, but as soon as I saw thumbnail and title I felt this video features Dr Grime

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.

We should use the z axis, use a little tunnel or bridge over the crossing points!

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

@GerSHAK

6 ай бұрын

+

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.

He looked so happy when he got to name the variables after (k)ilns and (s)torage units

Would love to see a sequel to this that includes more dimensions, or plots the scenario on the surface on a sphere or manifold.

Only true mathematicians are crazy enough to think of maths problems while working in life threatening situation .

@NitinSatendraRawat

11 ай бұрын

@@Holofractalius Totally agree .Thanks for suggesting.

@yt.personal.identification

11 ай бұрын

If you want the most efficient way to do a task, ask a lazy person to do it.

@I_Love_Learning

11 ай бұрын

@@Holofractalius Yep, if the lazu person is too lazu to use their brain, nothing will happen. You need to force them to do it!

@Cellottia

11 ай бұрын

​@@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...

@aitehs

4 ай бұрын

Hungary sided with nazis in ww2. He was manager of some sorts, I suppose

I love it when mathematicians make theoretical conjectures out of real life problems, rather than addressing the practical issue and fixing the rail switch mechanisms

@DonkoXI

11 ай бұрын

This is basically all we ever do. Even with math. We'll see a math problem, think about a pattern it has and formulate a new problem and get lost trying to answer that one instead.

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.

Well I timed that refresh perfectly...

@m3mefy

11 ай бұрын

Hi Squonk 😊

@DoctorKoffee

11 ай бұрын

same!

@drenz1523

11 ай бұрын

first comment?!

@asheep7797

11 ай бұрын

Wow.

@HoSza1

11 ай бұрын

what refresh?

Another fantastic Numberphile video. I am so glad I found this channel 6 years ago.

I was fixated on that spiffy travel schedule display on the wall the whole time.

In practical terms there would be a no crossing, just need to converge all tracks to a rotary platform.

The brick factory problem happens when I don't eat enough fibre

@Cellottia

11 ай бұрын

Or drink enough water? Have you tried magnesium supplements?!

Always a pleasure to see Dr. Grime!

This is a lot of work because the tracklayer did a shoddy job. the solution here isn't to break your brain with crossings, but to punish the tracklayer and make him do it again, but better.

@joleneonyoutube

11 ай бұрын

LOL I love this answer

Oh snap, Singing Banana is still at it, all these years later! I'm so happy about that :)

There’s a reason why railway intersections exist.

So computers chips are full of kilns, bricks, storage units and railway tracks? I've always suspected as much.

"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...""

It's incredible how I could listen to James "Weasley" Grime all day long

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.

Imagine he comes back to his boss like: i solved it but it only works in 27 dimensions

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.

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.

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.

@bonsoonlin

11 ай бұрын

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.

@beardedboulderer2609

11 ай бұрын

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.

@vorpal22

11 ай бұрын

@@beardedboulderer2609 Thanks for the heads up! I will absolutely check out that theorem.

@JohnDoe-ti2np

11 ай бұрын

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.

@mobius32

11 ай бұрын

@@beardedboulderer2609 came here to mention the Robertson-Seymour excluded minor theorem! Nice work.

It had to be a mathematician that when put in a forced labour camp thought "how can I optimise productivity for my captors?"

You’re kiln-ing me with these extremely fascinating videos!

Dude was forced into slave labor and still managed to turn it into a mathematical exercise, absolute legend

@rtpoe

11 ай бұрын

Using your brain like that helps keep you sane. There are PLENTY of stories from POWs about such distractions.

sounds like the book-page graph problem is a special case of the everything to everything version of this problem

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...

Another obvious solution is to utilize the fact we're working in three dimensions, and have a bridge over the track.

This reminds me of the Utilities Mug on Maths Gear.

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.

@ddognine

11 ай бұрын

True, PCB design is as much art as science and the more layers you have, the more expensive to manufacture.

@fieldrequired283

11 ай бұрын

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.

@chrisgriffith1573

11 ай бұрын

@@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!

@JasonMitchellofcompsci

11 ай бұрын

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.

@DonkoXI

11 ай бұрын

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.

I’d love to hear about more real world applications for this sort of thing! That circuit example was brilliant 😊

Excellent explanation thank you so much.

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.

Just connect all the storage units together and run each kiln to one storage unit in most cases then they can shuffle between units.

Now im interested in how they calculate the minimum amount of layers for computer chips :P

@victorcossio

11 ай бұрын

You work in 3D instead of 2D

@ricardorix73

11 ай бұрын

I think he means PCB's which can be multi-layered.

@FrankHarwald

11 ай бұрын

(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.)

@Lattamonsteri

11 ай бұрын

@@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

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

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?

James Grime with the first pose full of style sheesh

A new numberphile video with James Grime? Obviously a no-brainer to watch it immediately!

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.

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

James has the most clickbaity problems. He seems like the most interesting mathematician. Can't believe i've watched him for over 10 years.

That's a nice theoretical problem, fit for a mathematician. An engineer would think "Hmm, what can I change in the design of the trolley and the tracks, so that the darn thing doesn't flip each time it goes over a crossing?"

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’ll always watch and Like JAMES GRIMY videos

The realistic solution is to just reuse the tracks. You can just make a roundabout or go through another point.

@ColonelSandersLite

11 ай бұрын

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.

This reminds me of a scam school episode. The trick was if you draw a 3 to 3 connection it doesn't work without an intersection on a 2d surface but it does work on a bagel shape item

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.

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?

@TlalocTemporal

11 ай бұрын

That must be disallowed, otherwise all graphs could be solved with only one or two crossings. Just send everything into the rail vortex.

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.

taking notes for my next Factorio run

Wow, I was just reading about this problem yesterday!

The optimum solution to the 3+3 case is to build your brick factory on the surface of a coffee cup.

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

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.

With the stuff I'm working on I'm still frustrated graph theory hasn't been fully explained yet.

3 kilns and 3 storage units is a tradition around here. Grant Sanderson enters the the video with his coffee mug!

I love how they just casually have their channel's stats on the wall behind them

After Pal Turan was finished, all of the forced labour in that camp was done by a single Australian man (Futurama reference)

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.

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.

@bitsandbobs42

11 ай бұрын

There's only a couple of tracks that don't need to cross

@RamsesTheFourth

11 ай бұрын

Then the mathematicians would have nothing to eat!

I love this channel!

Hi idk if you’ll see this but thank you for your videos and keep up the great work

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.

@parthon

11 ай бұрын

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.

Pass through stations at each kiln and storage unit. One loop for all kilns, one loop for all storage units, two connections between the loops. Zero crossings.

3K,3S is no problem if you have a large enough coffee mug to build on.

This reminds me of a game called gPlanarity where you're given a graph and have to drag the nodes around to minimize the number of crossings.

Everything changes when we start thinking in 3 dimensions for this problem

Maybe that's where the expression 'bricking it' came from 😂

You know, it's kinda nice to see floor division outside of programming.

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.

Anything: (exist) Mathematics: "And that's a problem"

@asheep7797

6 ай бұрын

Bro's been watching Numberphile for at least the past 5 years.

Great problem, great background story

You can actually summon a nerd demon with that pentagram

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.

@MathPickle

11 ай бұрын

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."

Dr Grime my beloved

add sidings to each node, add switches where needed, you'll need 0 crossings. Crossings themselves can be nodes of their own for larger cases

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

Imagine a prisoner coming up to a guard and showing him a mathematical model to make the work camp more efficient.