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The
WaterGasandElectricity Puzzle by Gianni
A. Sarcone 
Solutions
for your logic and mechanical puzzles
"Dear
Archimedes Lab, if you have 3 houses and each need
to have water, gas and electricity connected, is
it possible to do so without crossing any lines?
Can you please post the solution? Thank you very
much!"  Gerald

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 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. 
PROBLEM
AND SOLUTION
Water,
Optical fiber, Light  a connection dilemma
Each year, we receive an extraordinary number of letters
regarding a classic route problem called “Water,
Gas and Electricity”. Since the puzzle is very
famous, we have chosen to present it in a new light.
Ok... We have laid on water, internet connection and
electricity from the utility suppliers W,
O, L to each of the 3 houses A, B and C without
any pipe crossing another (see fig. below). Take a
pencil and check if the work has been done properly!
See
the solution
Once
and for all... NO, it isn’t possible to connect
the buildings W, O, L to
each of the 3 houses WITHOUT intersecting a pipe.
If we differentiate with colors the relative connections
which start from the utility suppliers W, O and L,
we can see on the image that some houses (A and C,
see image below) are connected twice to the same
utility supplier!
Source: Puzzillusions,
Sarcone & Waeber, ISBN: 1844420647.
Simple
explanation
Why is it impossible to solve this puzzle
in 2 dimensions? Have a look at the diagram in fig. a below
and you will understand that 3 connections starting
from 2 utility suppliers will inevitably ENCLOSE one
of the houses preventing it from being connected to
at least one utility supplier (according to Jordan's
curve theorem, a loop or a closed curve will have
an inside and an outside no
matter how we stretch or curve our lines, as long as
they don't cross).
Alternative
solution 1
Nevertheless,
this puzzle is possible to solve by using subterfuge...
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 or a utility company and
then emerge from the building on the other side. In
fact, the wording of the puzzle is a bit imprecise
and doesn't forbid lines to go through the houses or
to use the third dimension!
Alternative
solution 2
Here is another neat way to solve it: reproduce the
puzzle on a paper sheet, then roll it up to form a
cylinder and add a paper strip to it as shown in fig. b.
The final image c shows how the puzzle should
appear and how the houses A, B and C are
finally connected to the utility suppliers.
Alternative
solution 3
There
is another neat and elegant way to topologically solve
the puzzle! To get a hint or to suggest a solution,
please, contact us. But
if you are impatient visit our FB
page to see the solution and click like
if you liked it!

Maths
behind the puzzle: Graph theory and examples
A planar
graph is a collection of points connected
by lines, that can be drawn on the plane in such a
way that its lines (called edges) intersect
only at their vertices (endpoints). In other words,
a planar graph, unlike the other complete
graphs, can be represented with no intersecting
edges. For instance, the graph in the example below
with 4 vertices (K_{4}) is planar
because if we move the vertex 4 through and beyond
the triangle 123, we can see that there are no more
edge intersections.
'The
Water Gas and Electricity problem' actually asks
whether the complete
graph K_{3,3} with two sets
of points, each point of which connecting three points
of the other set, is planar. If we perform the same
transformation to this graph, we notice that there
are always at least two edges that intersect, as
shown in the fig. e.
Euler
characteristic
Math enthusiasts can use the simple Euler
graph formula V  E + F =
2 (where V = number of Vertices, E =
number of Edges, and F = number of Faces)
to discuss and prove that this puzzle is unsolvable...
In our puzzle, the houses and utility suppliers together
represent the Vertices, and the Faces are the areas
inside a closed loop of Edges (this formula counts
the area outside the graph as one of the Faces). Important:
there can't be any Vertices in the middle of a Face.
If
we connect the utility suppliers to the houses we
obtain then 3 x 3 = 9 Edges (see graph K_{3,3} above),
because each of the 3 utility suppliers is connected
to 3 houses. We know that the boundary of every Face
is a closed loop of Edges, and we know that every
Edge goes between a house and a utility supplier
(there is no reason to go from a house to a utility
and back to the same house). That means the boundary
of a Face is made by at least 4 Edges (see fig. f opposite).
Now let’s use Euler’s formula to figure
out how many faces there are in the puzzle:
V  E + F =
2
F = 2 + E  V
F = 2 + 9
 6
F = 5 faces
Every Face has at least 4 Edges, so the number of Edges
in all the Faces is at least 4 x 5 = 20 Edges. This
counts each Edge twice, as every Edge is a boundary
for 2 Faces. So, the smallest number of Edges is 20
/ 2 = 10 Edges. We know, however, that there are only
9 Edges! This is a contradiction... Since nothing can
have 9 edges and 10 edges at the same time, drawing
a solution to the three utilities puzzle must be impossible.
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