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The Not So Famous Utilities Puzzle & Solution


You have three houses or squares drawn on paper and below them three smaller squares representing gas, water, and electric.  Can you draw a line and get each utility into every house (9) total lines, without crossing over any line?

Most common solution or should I say No Solution. 
Diagrams Are Below....

Without the drawing being 3D this puzzle cannot be done. Many people say it can but using the drawing above it cannot. The best result you may get is only eight proper connections. Other variations of the puzzle can be found below.  People who are much brighter than I can explain better so here is what some of them say about the puzzle.

 It is known as the Turan Brickyard Problem.  The formula says this let n = the number of houses and m = the number of utilities.  Then the minimum crossing number is known by the formula
 Cr(n,m) = [n/2] [(n-1)/2] [m/2] [(m-1)/2] where [x] denotes the integer part of x.
 So the minimum crossing number of Cr(3,3) = [1.5] [1] [1.5] [1] = 1 * 1 * 1* 1 = 1.
 So they can NOT be connected without crossing. M. Adair
The only way this can be done without the lines crossing is by allowing one of the lines (it doesn't matter which one) to enter a house then emerge from the house on the other side of the house when it will be able to enter another house without any other line having to be crossed.
       Keep on sending out the quizzes: you give a lot of people a whole heap of pleasure.
   
       Regards,           Neil Phillips
(This puzzle is a classic one which has no solution in
2D. However, if you place the items on a doughnut
shape in 3D you can solve it. )
regards
Lisa Belfield
Go to this website and scroll down:
 It looks like "Eight Is Enough."
 Bari Graves
Based on the statement that 'drawn on paper' implies that this is a 2D surface then one can NOT connect the 3 utilities to the three houses.  I am searching for a source for this but haven't found one.  It is related to the mathematical "Four Color Map Theorem" but I can't find the exact source.

Were you to change the wording and state that the surface was not 2D but say a 3D Sphere then the answer is yes.

-Mike

They answer is, you have to see things in a three dimensional kind of way, you
have to roll the piece of paper, which will enable you to re-route all three
gas, electric and water to each house, without having any of the lines touch
or cross each other

 

We did the "three houses, three utilities" question last year in
Math's class, and it has been proved to be impossible on a 2-D
representation due to the fact that there is an odd number of lines,
etc... If I had my math's book here I could explain it, but trust me -
it is impossible unless you made the representation 3-D, which is not
the question! Hopefully someone who has their math's book or similar
can explain the theory behind the question! ;-)
Hope I helped,
Jessi

Other variations

Align 3 triangles over 3 circles.
Connect each of the triangle to each of the circles.
Do not cross any lines.
Do not fold the paper.
Stay on one side of the paper.
Do not intersect any of the circles.
Do not rearrange the shapes.
You have 3 squares and 3 circles and you have to drawn a line from each circle to each square without overlapping them and you cant put the squares in the circles or the circles in the squares please help me thank you 
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