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's.
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.
(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
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
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
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
