[random7]

This is a twist on this puzzle "Somebody please help" posted by magicmike (also, this thread has a nice little pictured solution that compliments that puzzle: "Don't Cross The Lines")

Firstly, on a piece of paper, draw the three houses and utilities as boxes (or what ever shape you want) such that they are positioned as follows:

Now, connect each house (by drawing solid lines) to all three utilities, without crossing lines, and without going through houses/utilities (unlike in the previous puzzle). In this puzzle the houses/utilities are considered as nodes.

As usual, I will let you guys ponder for a day or two, before posting the answer in a spoiler. Please use spoilers, and yes, it is solvable. Enjoy

Edited August 8, 2008 by random7

[Guest]

temporarily locked

[Guest]

bump

[Guest]

I can't wait to see the answer on this one! I still don't think it's possible

[random7]

Ok... This thread isn't doing so well, so I'll post a hint...

The phrase "Firstly, on a piece of paper, draw the three houses and utilities..." is a key statement.

I'll probably be posting clues regularly, since I don't think this thread is going to improve in posts...

[Guest]

does this include folding the paper?

[random7]

does this include folding the paper?

You can if you want... But keep in mind that you have to draw solid lines on the paper (ie. not broken) ... (I'd be quite impressed if you found a solution this way though, so do try it)

[random7]

Drawing on a piece of paper is like drawing on the surface of a sphere, a cube, or any object that can be transformed (by stretching, bending, etc.) into a sphere.

The problem with this is that the "Three Utilities" problem is impossible to solve when it's presented on the surface of a sphere (ie. piece of paper) let alone a plane (which is sort of like a sphere with infinite radius).

The hint is in this question: "What else can I do to paper, that isn't folding it or stretching it (which will in turn transform it into something that isn't likened to a sphere)?"

I will post the answer about 5 hours from now (ie. when I get home).

[brainz]

Drawing on a piece of paper is like drawing on the surface of a sphere, a cube, or any object that can be transformed (by stretching, bending, etc.) into a sphere.

The problem with this is that the "Three Utilities" problem is impossible to solve when it's presented on the surface of a sphere (ie. piece of paper) let alone a plane (which is sort of like a sphere with infinite radius).

The hint is in this question: "What else can I do to paper, that isn't folding it or stretching it (which will in turn transform it into something that isn't likened to a sphere)?"

I will post the answer about 5 hours from now (ie. when I get home).

you can burn paper does that have anything to do with it?

[Guest]

Drawing on a piece of paper is like drawing on the surface of a sphere, a cube, or any object that can be transformed (by stretching, bending, etc.) into a sphere.

The problem with this is that the "Three Utilities" problem is impossible to solve when it's presented on the surface of a sphere (ie. piece of paper) let alone a plane (which is sort of like a sphere with infinite radius).

The hint is in this question: "What else can I do to paper, that isn't folding it or stretching it (which will in turn transform it into something that isn't likened to a sphere)?"

I will post the answer about 5 hours from now (ie. when I get home).

If you can use the backside of the paper, you can punch holes in it and solve it that way. But I consider this to be outside of the bounds of the original problem statement, which said two lines cannot cross. Just because the line is on the other side of the paper, doesn't mean that when you trace over the same area it isn't crossing. I could just as well layout a small piece of paper over my old track, and go across it that way.

So, unless you have another, trickier solution, I call shenanigans.

[brainz]

If you can use the backside of the paper, you can punch holes in it and solve it that way. But I consider this to be outside of the bounds of the original problem statement, which said two lines cannot cross. Just because the line is on the other side of the paper, doesn't mean that when you trace over the same area it isn't crossing. I could just as well layout a small piece of paper over my old track, and go across it that way.

So, unless you have another, trickier solution, I call shenanigans.

you mean something like this?

[random7]

If you can use the backside of the paper, you can punch holes in it and solve it that way. But I consider this to be outside of the bounds of the original problem statement, which said two lines cannot cross. Just because the line is on the other side of the paper, doesn't mean that when you trace over the same area it isn't crossing. I could just as well layout a small piece of paper over my old track, and go across it that way.

So, unless you have another, trickier solution, I call shenanigans.

1) Yep you've got it (the hole part)... You actually only need one hole, but there isn't anything wrong with using multiple holes. Also, the below picture is only one solution of many configurations.

2) The lines don't actually cross in the 2D space (as will be explained below), even though when looking at them at various angles in 3D space they seem to.

3) Laying another piece of paper across the original piece of paper counts at a break in the line, since the original piece of paper is it's own 2nd dimensional space.

Ok here goes with the answer:

When drawing on a piece of paper, it is likened unto drawing on the surface of a sphere (or more relevant in this case, a cube), since a piece of paper is a 3D closed surface object. No matter how you rip (by this I mean from the edges) or bend a piece of paper, it will always have the same attributes as drawing on a sphere. But! By poking/cutting a hole in the paper, the piece of paper now becomes likened to drawing on the surface of a torus, allowing the puzzle to be solved. The red broken line represents the line being drawn on the reverse side of the paper (and it's actually a smooth line as well, that is, it has no sharp bends relative to the 2D space).

Here is a picture by Dr. Math that shows the solution drawn on a torus:

By specifying that this puzzle is to be done on paper, is the key difference to the related puzzle. If I were to ask for you to solve it on a 2D plane (or even a sphere), it would be impossible, unless I were to change the restrictions.

Another note to clarify that this is still working in the 2nd dimension: if Homer Simpson were to live in the 2nd dimensional surface of a torus, the problem would seem impossible. When it is solved however, he would see that all of the pipes and powerlines did not come into contact, and would conclude that it must be because of some phenomena he might call a "wormhole" (which he cannot see). In the 3rd dimension, the space that we live in, there would be a puzzle that is equivalent to this one, which would seem to be unsolvable. But, if it were possible for us to see in the 4th dimension, we might be able to see how our 3rd dimensional space curves and bends in the 4th dimension, allowing us to know where to route the pipes and powerlines (ie. through things like wormholes).

All of that explanation is an attempt to show that a surface is still a 2D space, just like our 3D space is still a 3D space, even though it might not be linear (linear is likened to describing a 2D surface as "flat"). And if the lines aren't crossing in the 2D space, they definitely cannot be crossing in 3D space (even though they look like they are, on curtain angles, since our vision is pretty much restricted to 2D -- or simulated 3D).

If you have any questions on this, I would be glad to answer them.

[Guest]

Drawing on a piece of paper is like drawing on the surface of a sphere, a cube, or any object that can be transformed (by stretching, bending, etc.) into a sphere.

The problem with this is that the "Three Utilities" problem is impossible to solve when it's presented on the surface of a sphere (ie. piece of paper) let alone a plane (which is sort of like a sphere with infinite radius).

The hint is in this question: "What else can I do to paper, that isn't folding it or stretching it (which will in turn transform it into something that isn't likened to a sphere)?"

I will post the answer about 5 hours from now (ie. when I get home).

Make it into a mobius strip. That'll untwist the crossover.

[random7]

Make it into a mobius strip. That'll untwist the crossover.

Sounds like it might work... Have you actually solved it on a mobius strip?

Edited August 11, 2008 by random7

[Guest]

Sounds like it might work... Have you actually solved it on a mobius strip?

Here's a kind of drawing of it - first the solution with one crossover in the plane, and second using a mobius strip - the half twist in the strip ensures that the two lines do not cross round the back!

Feel free to make it yourself... I needed to to convince myself it worked!

[Guest]

Here's a kind of drawing of it - first the solution with one crossover in the plane, and second using a mobius strip - the half twist in the strip ensures that the two lines do not cross round the back!

Feel free to make it yourself... I needed to to convince myself it worked!

Umm... embarrasment. Doesn't work (at least not with that drawing) the lines go through the twist twice and we end up connecting house to house and utility to utility.

However.. it may be feasible. IIRC a mobius strip and a torus both have a euler characteristic of zero. I'll get scribbling on my mobius strip.

[random7]

Umm... embarrasment. Doesn't work (at least not with that drawing) the lines go through the twist twice and we end up connecting house to house and utility to utility.

However.. it may be feasible. IIRC a mobius strip and a torus both have a euler characteristic of zero. I'll get scribbling on my mobius strip.

HA! It works! That is awesome... I'm excited because I like Mobius strips... And it even (kind of) fits with the first post's conditions... Like not many people have a solid Mobius strip lying around... Which would count as a piece of paper (sort of) ... But anyway, it is still an another awesome solution to the unsolvable "Three Utilities" puzzle

In response to your post... You just have to go around the edge of the Mobius strip, that is, similar to what I have done in my torus solution (see picture in Answer spoiler).

Also, that Euler characteristic thing sounds like it will pop up in my "Differential Equations" math subject I'm doing at uni.

[Guest]

Umm... embarrasment. Doesn't work (at least not with that drawing) the lines go through the twist twice and we end up connecting house to house and utility to utility.

However.. it may be feasible. IIRC a mobius strip and a torus both have a euler characteristic of zero. I'll get scribbling on my mobius strip.

I think if one is considering a piece of paper a topological sphere, then one must be able to go over the top of an edge. And so, when you construct your moebius strip, you could just send the line over an edge. It may not be as elegant as before, but it still is a very original solution, I think.

[Guest]

HA! It works! That is awesome... I'm excited because I like Mobius strips... And it even (kind of) fits with the first post's conditions... Like not many people have a solid Mobius strip lying around... Which would count as a piece of paper (sort of)

... But anyway, it is still an another awesome solution to the unsolvable "Three Utilities" puzzle

In response to your post... You just have to go around the edge of the Mobius strip, that is, similar to what I have done in my torus solution (see picture in Answer spoiler).

Also, that Euler characteristic thing sounds like it will pop up in my "Differential Equations" math subject I'm doing at uni.

Yes... but I was hoping to avoid going over the edge. It might answer the original question, but its not as elegant as I was hoping.

Euler characteristic is a topology thing. And I think is a red herring... it seems to be the hole in the torus that is the important factor.

It'd probably work on a Klein bottle...