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Connect every house with every well without the lines intersecting.

enter image description here

I am not sure if this puzzle has a solution. I have been puzzled by it for a long long time. An old man from my village mentioned this puzzle to me 12 years ago. I have never been able to solve it, I'm hoping someone here can.

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    $\begingroup$ nomachetejuggling.com/2011/10/29/… $\endgroup$
    – Anon
    Jul 1, 2015 at 14:08
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    $\begingroup$ @Anon well since houses and wells are 3D things, you can always dig some tunnels or build bridges. $\endgroup$
    – shy
    Jul 1, 2015 at 14:15
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    $\begingroup$ This is a well-known puzzle and I was sure it must be a duplicate, but I couldn't find it anywhere on the site already. Have an upvote :-) $\endgroup$ Jul 1, 2015 at 15:43
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    $\begingroup$ I, too, assumed that this puzzle must already be on here somewhere. I don't think that this post has yet become a more accessible or complete answer than one of the top google hits, though. Indeed, an answer already links to that. I'm waiting to see how much more we can expand on that site's information. $\endgroup$ Jul 1, 2015 at 16:03
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    $\begingroup$ I think that the puzzle should be edited to specify that each hous must be connected to each well using a different pipe and that pipes should not cross the wells. Most of the solutions proposed are really creative but, IMHO, on purpose misunderstanding the sense of the puzzle... $\endgroup$
    – Hunter
    Jul 2, 2015 at 6:49

11 Answers 11

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This is not impossible.

While the graph theory is correct, it's also not relevant. The wells in question aren't vertices, and so we can connect multiple edges to a single well without them overlapping.

Here is a possible solution:

Each house has a pipe that connects to all 3 wells (I've drawn on openings for each pipe in each well). The pipes do not overlap.
houses and wells

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  • $\begingroup$ Actually instead of 1 pipe you can use 3 pipes that are very closely together, and your solution still works. The reason it works is because you allow pipes to cross the wells and not originate from a single point. You can also solve this by replacing the wells with vertices, but allow pipes to cross the houses. (Just turn your current solution upside down and you have that) $\endgroup$
    – Dorus
    Jul 1, 2015 at 19:26
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    $\begingroup$ Another solution is to share pipes, since it doesn't say you can't do that you can connect all three wells to house 1, and then simply connect house 2 and house 3 to house 1's supply. $\endgroup$ Jul 1, 2015 at 21:25
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Probably a bit simpler than what you were looking for. Looking at the other answers, I feel like there might be some details missing from your question.

my solution

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This link explains why this puzzle is unsolvable:

It also suggests a clever solution in "3 dimensions"
(actually, a 2D solution transfomed into a 3D one).

3D Solution

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    $\begingroup$ If one uses 3D it's not necessary to use a torus, because any intersection can be avoided by placing a pipe over/below another one (digging it in deeper) $\endgroup$
    – Voitcus
    Jul 2, 2015 at 5:36
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    $\begingroup$ For the awesomely nice picture (which I understand you designed and drew yourself just to make the answer be nice, right?) I think you deserve +1. $\endgroup$ Jul 3, 2015 at 10:09
  • $\begingroup$ Its line a "particle accelerator". :D Be careful. :). BTW. very nice answer $\endgroup$ Jul 3, 2015 at 11:02
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    $\begingroup$ @Voitcus: It's funny how times change. When I went to school (1960s) a solution involving pipes over/below was regarded as "cheating", since the purpose of the problem was to demonstrate what isn't possible on a flat plane but is possible on a torus. Topology 101, I suppose. I vaguely recall there might have been a SciAm article about it. $\endgroup$
    – MartynA
    Jul 3, 2015 at 19:26
  • $\begingroup$ However, I like the torus solution and +1'd it $\endgroup$
    – Voitcus
    Jul 4, 2015 at 14:48
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Option 1:
Ask a mathematician to explain why the complete bipartite $K_{3,3}$ graph is non planar.

Option 2:
Put this drawing and a stack of cash in an envelope. Deliver the envelope to a competent engineer.

enter image description here

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    $\begingroup$ This is hilarious, but not really an answer ... $\endgroup$ Jul 1, 2015 at 15:41
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    $\begingroup$ True...but it is a solution $\endgroup$
    – Anon
    Jul 1, 2015 at 15:42
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    $\begingroup$ @randal'thor The question doesn't state that the pipes cannot cross one another. Just that they can't intersect. $\endgroup$ Jul 1, 2015 at 20:40
  • $\begingroup$ @DavidRicherby If you want to get technical, the puzzle doesn't even mention pipes, just lines. And those lines definitely intersect... $\endgroup$
    – Cain
    Jul 1, 2015 at 21:56
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    $\begingroup$ @Cain It's a 2-d representation of a 3-d solution in which the lines don't intersect. $\endgroup$ Jul 1, 2015 at 22:32
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This is impossible

Assume 2 houses are fully connected to 3 wells. There must be an 'inner' well which is surrounded by the connections to the other 2 wells.

For the third house to be connected to the inner well, it must be in one of the spaces between the inner well and one of the outer ones. But this means it is seperated from the other outer well. There is thus no way for the third house to join all three wells.

Edit: this answer assumes that lines may not cross a well/house en route to another.

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    $\begingroup$ You're assuming that the pipes can't cross under or over each other. That's normally stated as a condition but this particular telling of the question doesn't mention it. $\endgroup$ Jul 1, 2015 at 20:39
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    $\begingroup$ Assume 2 houses are fully connected to 3 wells. There must be an 'inner' well which is surrounded by the connections to the other 2 wells. That part must be proved too. $\endgroup$
    – Cthulhu
    Jul 2, 2015 at 10:55
  • $\begingroup$ But this means it is seperated from the other outer well. There is thus no way for the third house to join all three wells this is not rigorous as well. $\endgroup$
    – Cthulhu
    Jul 2, 2015 at 10:58
  • $\begingroup$ @Cthulu You're right, it's not rigorous. This is puzzling, not The Journal of Mathematical Tedium. $\endgroup$ Jul 2, 2015 at 11:04
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Not possible

According to graph theory it is the smallest non plannar graph with minimum number of lines.

In graph theory, a planar graph is a graph that can be embedded in the plane. This graph is commonly called as Utility graph or $K_{3,3}$.

There is a lot of interesting problems which can solve(or can prove not solvable :p ) using Graph theory.

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enter image description here

The blue is going through the house.

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    $\begingroup$ Not sure why this is getting downvotes ... the question doesn't actually say the lines can't go through the houses! $\endgroup$ Jul 4, 2015 at 9:26
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Not possible

House 1 and House 2, when connected to Well 1 and Well 3 will form a closed shape around Well 2. House 3 will NOT be able to access one of the wells, regardless of where you put it.

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Hi,

Is this not good enough? :/

Edit: Missing a pipe, damn it. I tried adding it but I kept ending up requiring some kind of magic

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  • $\begingroup$ I do think that's the first time I've seen anyone solve the puzzle like this, without having to draw singular pipes to each house or go into three dimentions. Great job. $\endgroup$
    – Nefer007
    Jul 3, 2015 at 14:29
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    $\begingroup$ You missed one: House 2 is not connected to Well 3. $\endgroup$
    – mmking
    Jul 3, 2015 at 14:30
  • $\begingroup$ ARGGHHH DAMN IT! $\endgroup$ Jul 3, 2015 at 14:34
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    $\begingroup$ Though you missed one, you perfectly show why it's impossible to do xD $\endgroup$ Jul 3, 2015 at 14:42
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    $\begingroup$ Lets just say I intended to show why its impossible.. to illustrate the point. (surely that deserves some up votes?) $\endgroup$ Jul 3, 2015 at 14:44
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An old man from my village mentioned this puzzle to me 12 years ago. I have never been able to solve it, I'm hoping someone here can.

If the old man ask you to do this on a paper page, there is a solution with this small "legit hack":

just make a hole in the paper, and draw a line through it, then continue the line on the other side and comme back through another hole. enter image description here

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I am not sure this is the correct answer, but this is the best way that I found to answer this puzzle:

houses and wells interconnected without  the lines intersecting

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