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Flatland Problem

By Onvacation follow Onvacation   2021 Aug 21, 11:47am 761 views   78 comments   watch   nsfw   quote   share      


My ninth grade math teacher gave us this problem


The problem: Connect each house with a line from the gas, water, and electricity companies without crossing any lines. You can put the houses and utility companies anywhere you want. You can use both sides of the paper; when your line gets to the edge just continue over to the other side.

My math teacher said if we solved it he would give us an A for the year. On top of that he said if we solved it in high school he would talk to our math teacher and get us an A.

I spent a lot of time the next 4 years attempting but never solving this problem.



It seemed there was always one connection that could not be completed without violating the rules.

Has anyone seen this? Has anyone else solved it?

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39   Tenpoundbass   ignore (16)   2021 Aug 22, 11:48am     ↓ dislike (0)   quote   flag        

Use a hole puncher, and punch a hole above H2m run the H20 line going to the H3, into the hole to come up the other side, as it is now on the right.
Run H1 electric on the front of the page, by routing it to the far left side of the page, and running it up to the left of the hole, and up to H1.
40   WineHorror1   ignore (2)   2021 Aug 22, 12:03pm     ↓ dislike (0)   quote   flag        

When do we get to see the solution?
41   richwicks   ignore (2)   2021 Aug 22, 12:05pm     ↓ dislike (0)   quote   flag        

qroproton says

If you remove the line from House 2 to Gas, then redraw it going up, around the other side of the page, and then up to Gas, that allows you to run a line from House 3 to the edge of the page, then across the backside emerging on the LHS of the page so that you can connect to Water.


Doesn't work, electric and water lines cross on the back of the paper.

This has no solution I don't think. I spent about an hour simplifying it in my head last night.

It seems like being able to go over to the other side of the paper would be an advantage, but them I realized that's the same constrains you have on a sphere. A sphere is really no different than a flat surface - take a sphere, poke one hole in it, then stretch that hole out to a square. All a flat sheet is is a sphere, missing a single point.

Doesn't matter where you place the homes and utilities either.
42   Karloff   ignore (0)   2021 Aug 22, 12:17pm     ↓ dislike (0)   quote   flag        

You cannot solve this if you are restricted to 2 dimensions. Hence the name "Flatland".

It's an interesting conundrum, and I think it's best aspect is that it points out that many problems cannot be solved if you limit yourself to restrictions put upon you by others (including nature) and that thinking outside of the box is what allows the impossible to become possible.

As an example, while faster-than-light travel may not be physically possible as we know it, the same effect, that is, to travel great distances in reasonable time, might be achievable through other means. eg. Wormholes.
43   Automan Empire   ignore (1)   2021 Aug 22, 12:52pm     ↓ dislike (0)   quote   flag        

Onvacation says
Hint: You need to use both sides of the paper, with a twist.


You can just tell us, no need to be a Mobius Dick about it. ☜(゚ヮ゚☜) This takes us out of two dimensions however, as does the solution of projecting it onto a torus.


Tenpoundbass says
Are you allowed to put holes in the paper too, so the route can go to the back of the paper via the hole


Everything but the simplest 555 timer or add-2-caps-and-a-pot amplifier chip circuits use 2 sided circuit boards with plated-through vias or holes. The first single sided circuit boards that replaced discrete components connected point to point in 3 dimensions needed wire jumpers over the component side to solve this problem. Board architecture for compactness sometimes has sub-boards jumpered in perpendicularly or in a parallel plane, often using a flexible PCB and headers for the jumper itself. Then there are boards designed for automated parts placement all in lines; good luck trying to intuit the schematic by looking at one of these!
44   Onvacation   ignore (7)   2021 Aug 22, 1:20pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
It's pointless mental exercises like these that have dumbed down two generations. It seems all of the cheat answers are more accepted, than how things are actually done in practice.

I disagree. It's problems like this that make youngsters think. Even though the current culture discourages critical thinking over "Googling" we need critical thinkers to solve todays. and tomorrows, problems.

The problem introduces graph theory for which this particular problem has no planar solution. The answer, at least the one I discovered, was inspired by topology.

It is certainly possible to have a great life with no formal education but in good schools with good professors we learn of problems and solutions that scientists and mathematicians have worked on over the eons. Like Newton said, "I stand on the shoulders of giants."

This is not a practical problem. It's a "thought exercise". It has constraints. There are probably more than one solutions but the solutions offered so far in this thread have violated the constraints.
45   Onvacation   ignore (7)   2021 Aug 22, 1:22pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
Use a hole puncher, and punch a hole above H2m run the H20 line going to the H3, into the hole to come up the other side, as it is now on the right.
Run H1 electric on the front of the page, by routing it to the far left side of the page, and running it up to the left of the hole, and up to H1.

Now that is "thinking outside of the box". Us ninth graders tried that one as well. Mr. Schultz, our math teacher, said, "Nope."
46   Onvacation   ignore (7)   2021 Aug 22, 1:28pm     ↓ dislike (0)   quote   flag        

richwicks says

This has no solution I don't think.

No planar solution.Automan Empire says


You can just tell us, no need to be a Mobius Dick about it. ☜(゚ヮ゚☜) This takes us out of two dimensions however, as does the solution of projecting it onto a torus.

Automan got it!
47   Onvacation   ignore (7)   2021 Aug 22, 1:33pm     ↓ dislike (0)   quote   flag        

Thanks for playing all of you that did.
48   Tenpoundbass   ignore (16)   2021 Aug 22, 1:46pm     ↓ dislike (0)   quote   flag        

Onvacation says
Now that is "thinking outside of the box". Us ninth graders tried that one as well. Mr. Schultz, our math teacher, said, "Nope."


Sounds like he was just favorizing his solution.

Onvacation says
No planar solution.Automan Empire says


You can just tell us, no need to be a Mobius Dick about it. ☜(゚ヮ゚☜) This takes us out of two dimensions however, as does the solution of projecting it onto a torus.

Automan got it!


If you can project it onto another dimension, and that is accepted as a valid solution. Then all solutions are valid. As it's only a thought exercise, with no one true answer.

Take this problem for instance, there is no practical application for this, then it's just a factoid nothing more than a trivial notation.

https://www.youtube.com/watch?v=094y1Z2wpJg
49   Onvacation   ignore (7)   2021 Aug 22, 2:04pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
If you can project it onto another dimension, and that is accepted as a valid solution. Then all solutions are valid. As it's only a thought exercise, with no one true answer.

Only solutions that obey the constraints are correct. one piece of paper, front and back, no utility lines cross, all houses get all utilities.
50   mell   ignore (6)   2021 Aug 22, 2:15pm     ↓ dislike (0)   quote   flag        

The torus was one of the suggested solutions on the page you linked earlier as well though, iirc.
51   Onvacation   ignore (7)   2021 Aug 22, 2:24pm     ↓ dislike (0)   quote   flag        

I don't think you can easily, if at all, create a torus from a single piece of paper. I guess you could, but that is not the moebius solution I came up with.
52   Onvacation   ignore (7)   2021 Aug 22, 2:30pm     ↓ dislike (0)   quote   flag        

Funny story.

I looked up Mr. Schultz when I discovered the solution to this problem. At first he was skeptical, but agreed that I had found the solution after I explained it to him.

I asked him if he was still teaching. He replied that he was now a construction manager as he got tired of dealing with the educational bureaucracy and low pay.
53   Tenpoundbass   ignore (16)   2021 Aug 22, 3:37pm     ↓ dislike (0)   quote   flag        

Onvacation says
Only solutions that obey the constraints are correct. one piece of paper, front and back, no utility lines cross, all houses get all utilities.


Going through the houses did not violate that principle and having a hole punched on the page didn't violate that principle.

Punching a hole in the page is the toroidal solution IMO, as you can't take a sheet of paper and create a toroid without cutting away everything that is not the toroid.
But in reality all you need is a 2D donut shape. With the center cut out. Then at that point having the edges rounded or left square is semantics.
So if you're allowed to trim away paper for a toroidal shape, but not just punching a hole in a flat paper to achieve the same outcome. Then it's just favoritism of the accepted answer.

I find in Computer programming there are more than one way to achieve the same outcome. None of them are the right way, or the wrong way. But only by how they are used in context to the over all project.
54   Onvacation   ignore (7)   2021 Aug 22, 3:43pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
you can't take a sheet of paper and create a toroid without cutting away everything that is not the toroid.

Roll it up into a tube and staple the ends together.

Can this problem be solved on a donut? I never tried it.
55   Onvacation   ignore (7)   2021 Aug 22, 3:47pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
I find in Computer programming there are more than one way to achieve the same outcome.

And once you find a workable solution you often stop finding a better solution, no matter how crude and inefficient your solution is.
56   Tenpoundbass   ignore (16)   2021 Aug 22, 3:53pm     ↓ dislike (0)   quote   flag        

Automan Empire says
Everything but the simplest 555 timer or add-2-caps-and-a-pot amplifier chip circuits use 2 sided circuit boards with plated-through vias or holes. The first single sided circuit boards that replaced discrete components connected point to point in 3 dimensions needed wire jumpers over the component side to solve this problem. Board architecture for compactness sometimes has sub-boards jumpered in perpendicularly or in a parallel plane, often using a flexible PCB and headers for the jumper itself. Then there are boards designed for automated parts placement all in lines; good luck trying to intuit the schematic by looking at one of these!

Yeah I have a Randall ProTube 1000 II that has and ugly single sided board, and is jumpered with ribbon cables and molex connectors.
I want to point to point the amp. My Amp tech buddy put it on his bench, he is an authorized HiWatt manufacturer. He gave it back and said it looked like it was made with a bunch of surplus components that they had. So they designed the circuit around what they had on hand, and not the other way around.
I love the sound of it, but Caps from the 80's and some of the other components aren't as spry as they were when they were built in the late 80's.
I think he is just a gear snob and didn't want to be bothered with having to redesign the circuit to achieve the desired voltage on the plate and output, I would have been looking for. Or thought I couldn't afford him doing so.
It's not like I'm looking for a Fender Baseman clone circuit, or I would just buy the kit.
57   Tenpoundbass   ignore (16)   2021 Aug 22, 4:03pm     ↓ dislike (0)   quote   flag        

Onvacation says
And once you find a workable solution you often stop finding a better solution, no matter how crude and inefficient your solution is.


That is how you become a dinosaur. I have several solutions for almost every problem, as the solution evolves as the use case reveals itself. A simple quick and dirty solution, just allows you to concentrate on more important areas of concern. Then as the project evolves, more elegant solutions evolve. Then those become obsolete, as the complex object, pass through and to objects and routines, may reveal speed or resource issues. I'm always refining methods and ways I do things. I value my collection of bad ideas more than my collection of great ideas. As my bad ideas collection is an invaluable tool when you get enough experience behind it. My Code end up always deploying bug free, while collogues and other teams I warned about the pitfalls of the routes they were taking. Only to get to give them a Trump smile when someone else tells them, "He told you so!".

I owe all of my coding efficiency to my collection of very bad ideas. If you know all of the stupid shit you shouldn't do, then you're only left with all of the sensible options.
58   Onvacation   ignore (7)   2021 Aug 22, 4:15pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says
If you know all of the stupid shit you shouldn't do, then you're only left with all of the sensible options.

Theoretically.
59   Onvacation   ignore (7)   2021 Aug 22, 5:43pm     ↓ dislike (0)   quote   flag        

We tried this one



Schultz said Nope.
60   richwicks   ignore (2)   2021 Aug 22, 7:11pm     ↓ dislike (0)   quote   flag        

Onvacation says
Tenpoundbass says
If you can project it onto another dimension, and that is accepted as a valid solution. Then all solutions are valid. As it's only a thought exercise, with no one true answer.

Only solutions that obey the constraints are correct. one piece of paper, front and back, no utility lines cross, all houses get all utilities.


Demonstrate it.

Going to the back of the paper does not solve the problem. Lines cross on the back of the paper.

I contend there is no solution, but it's a good problem for students. Some problems can't be solved. That's a lesson a lot of people never learn.
61   Onvacation   ignore (7)   2021 Aug 22, 7:22pm     ↓ dislike (0)   quote   flag        

richwicks says
I contend there is no solution, but it's a good problem for students. Some problems can't be solved. That's a lesson a lot of people never learn.

No planar solution.

Moebius could do it. One piece of paper with a twist. It turns a two sided plane into a geometric shape with one side and one edge.

I agree that some problems can't be solved. Some problems we have yet to prove unsolvable.

@richwicks Looking forward to meeting up with you in person again one of these days.
62   richwicks   ignore (2)   2021 Aug 22, 7:32pm     ↓ dislike (0)   quote   flag        

Onvacation says
richwicks says
I contend there is no solution, but it's a good problem for students. Some problems can't be solved. That's a lesson a lot of people never learn.

No planar solution.

Moebius could do it. One piece of paper with a twist. It turns a two sided plane into a geometric shape with one side and one edge.

I agree that some problems can't be solved. Some problems we have yet to prove unsolvable.

@richwicks Looking forward to meeting up with you in person again one of these days.


We'll meet up. I'm curious as to the divergence between my generation and your generation, and generation Y.

I think the "generation gap" is about how we've been propagandized differently. I THINK I'm outside of that sphere however, I've thought that before to realize I was not. The generation gap is imposed upon us, top down, it's not natural.
63   Patrick   ignore (1)   2021 Aug 22, 10:37pm     ↓ dislike (0)   quote   flag        

Onvacation says
@richwicks Looking forward to meeting up with you in person again one of these days.



I'd be up for that too.
64   GreaterNYCDude   ignore (0)   2021 Aug 23, 6:20am     ↓ dislike (0)   quote   flag        

Patrick says
Onvacation says
@richwicks Looking forward to meeting up with you in person again one of these days.



I'd be up for that too.

I think we're long overdue for some sort of PatNet meetup.
65   Fortwaynemobile   ignore (3)   2021 Aug 23, 6:44am     ↓ dislike (0)   quote   flag        

GreaterNYCDude says
Patrick says
Onvacation says
@richwicks Looking forward to meeting up with you in person again one of these days.



I'd be up for that too.

I think we're long overdue for some sort of PatNet meetup.


We need some closer to LA.
66   komputodo   ignore (3)   2021 Aug 23, 9:44am     ↓ dislike (0)   quote   flag        

well if we are looking at the theoretical solutions, why not sett up the electricity with a wireless transmission system? lol
67   komputodo   ignore (3)   2021 Aug 23, 9:46am     ↓ dislike (0)   quote   flag        

for all of the gang that likes puzzles, have you ever tried to explain the MONTY HALL PROBLEM to anyone? It can get frustrating and hilarious.
68   mell   ignore (6)   2021 Aug 23, 9:53am     ↓ dislike (0)   quote   flag        

komputodo says
for all of the gang that likes puzzles, have you ever tried to explain the MONTY HALL PROBLEM to anyone? It can get frustrating and hilarious.


It's still my favorite one because it took me a while to grapple - of course the math clearly shows the 2/3 to 1/3 probability ratio after one door has been opened (by the game master in the know that the prize is not behind the door they chose top open) but the brain has a hard time to accept it. Many people won't believe it, but it's easier explained if you start with 1000 doors.
69   Tenpoundbass   ignore (16)   2021 Aug 23, 10:59am     ↓ dislike (0)   quote   flag        

richwicks says
Onvacation says
Tenpoundbass says
If you can project it onto another dimension, and that is accepted as a valid solution. Then all solutions are valid. As it's only a thought exercise, with no one true answer.

Only solutions that obey the constraints are correct. one piece of paper, front and back, no utility lines cross, all houses get all utilities.


Demonstrate it.



70   Tenpoundbass   ignore (16)   2021 Aug 23, 2:08pm     ↓ dislike (0)   quote   flag        

Tenpoundbass says


Don't leave me hanging, does this solution conform to the requirements?
71   Onvacation   ignore (7)   2021 Aug 23, 2:22pm     ↓ dislike (0)   quote   flag        

One of Mr. Schultz's constraints was you could not go through the paper.

From: https://www.archimedes-lab.org/How_to_Solve/Water_gas.html

*Category:* Topological graph theory.
*Name:* Water Gas and Electricity puzzle, Three Utilities puzzle, or Three Cottage problem.
*Material:* Pencil, piece of paper.
*Configuration:* There are three houses (or squares) drawn on paper and below them three smaller squares [or circles] representing gas, water, and electricity suppliers.
*Aim of the game:* Draw lines to get each utility into every house, without crossing over any line.
*Origin of the puzzle:* Unknown. Sam Loyd claimed that he invented this recreational math problem about 1903. But this puzzle is MUCH older than electric lighting or even gas, Loyd most probably modified a previously existing puzzle.
*Editor's notice:* This is a pure abstract mathematical puzzle that imposes constraints that would not be issues in a practical engineering scenario... As such, this puzzle CANNOT be solved.

The only way to solve this problem that I know of is to twist the paper into a moebius strip. Technically doing this turns the two dimensional paper into a three dimensional geometric shape, albeit a strange one with one side and one edge.
72   Onvacation   ignore (7)   2021 Aug 23, 2:28pm     ↓ dislike (0)   quote   flag        

I remembered this problem because an easy way to create graphs with Excel came to me in a dream.

The solution to the "water/gas/electricity" problem came to me in a dream after I studied graph theory and topology when I was a math major in college.

Dreams are cool!
73   komputodo   ignore (3)   2021 Aug 23, 10:28pm     ↓ dislike (0)   quote   flag        

Onvacation says
*Editor's notice:* This is a pure abstract mathematical puzzle that imposes constraints that would not be issues in a practical engineering scenario... As such, this puzzle CANNOT be solved.

Then why set it up as 3 houses with gas, elec, and water meters? why not just 3 circles and three squares?
74   komputodo   ignore (3)   2021 Aug 23, 10:30pm     ↓ dislike (0)   quote   flag        

mell says
It's still my favorite one because it took me a while to grapple - of course the math clearly shows the 2/3 to 1/3 probability ratio after one door has been opened (by the game master in the know that the prize is not behind the door they chose top open) but the brain has a hard time to accept it. Many people won't believe it, but it's easier explained if you start with 1000 doors.

Would the odds change if the player instead of Monty opened the door(s) randomly with 1000 doors and just by sheer coincidence just happened to leave 1 winner and 1 loser?
75   Onvacation   ignore (7)   2021 Aug 23, 10:36pm     ↓ dislike (0)   quote   flag        

komputodo says
Then why set it up as 3 houses with gas, elec, and water meters? why not just 3 circles and three squares?

For the same reason they don't explicitly teach second graders negative numbers. Graph theory is a little too abstract for ninth graders.

Math is often years ahead of technology, sometimes centuries. If E=MC^2, M=E/C^2; try wrapping your brain around that. The mathematical field of topology explains wormholes and theoretically explains the possibility of faster than light travel, though I don't even pretend to understand how.
76   SunnyvaleCA   ignore (1)   2021 Aug 23, 10:52pm     ↓ dislike (0)   quote   flag        

I once twisted the insides of a floppy disk into a Möbius strip and hung it on my cubical wall with the label: single-sided floppy disk. That got a chuckle from co-workers.
77   Tenpoundbass   ignore (16)   2021 Aug 24, 9:32am     ↓ dislike (0)   quote   flag        

Onvacation says
The mathematical field of topology explains wormholes and theoretically explains the possibility of faster than light travel, though I don't even pretend to understand how.


That's where math fails the Eggheads. Just because math fits a scenario, sometimes it is completely useless outside of those bounds.
A great example is Time. Out depiction of time through a math system of 12 values with 60 segments, parsed in 24 hour dimensions, spread out over 365 Days, among many time zones, has absolutely no value anywhere else in the cosmos, other than to explain our planets rotation around the sun. And it's sloppy at best, and has to be reworked periodically, with subtractions here and there.

It's possible that a wormhole only works on paper, and has no value in the Universe.
78   mell   ignore (6)   2021 Aug 24, 9:34am     ↓ dislike (0)   quote   flag        

Onvacation says
komputodo says
Then why set it up as 3 houses with gas, elec, and water meters? why not just 3 circles and three squares?

For the same reason they don't explicitly teach second graders negative numbers. Graph theory is a little too abstract for ninth graders.

Math is often years ahead of technology, sometimes centuries. If E=MC^2, M=E/C^2; try wrapping your brain around that. The mathematical field of topology explains wormholes and theoretically explains the possibility of faster than light travel, though I don't even pretend to understand how.


There are other ways to achieve faster than light travel that are less esoteric such as riding inside a gravitational wave while having your own propulsion adding to the total speed. Just sayin ;)

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