5
room puzzle variants

diagram
with 16 openings/doors

3D
rendering of the puzzle

This
old popular puzzle, called “Five Room House
puzzle” (also known as “Walls
and Lines puzzle”, or “Cross
the Network puzzle”), is canonically represented
as a rectangular diagram divided into five rooms, as
shown opposite. The object of the puzzle is to draw a
continuous path through the walls of all 5 rooms, without
going through any wall twice, and without crossing any
path. The path can, of course, end in any room, not necessarily
in the room from where it started. Some puzzle diagrams
represent the rooms with openings supposed to be doors.
In this instance, the challenge is to visit every room
of the apartment by walking through every door exactly
once.
Requirements
for solvability
Whether starting and ending in the same room, or starting
in one room and ending in another one, every other
room of the diagram/apartment must have an even number
of doors... That is, pair(s) of ‘in’ and ‘out’ doors (as
doors CANNOT be used TWICE, we have then to use an
even number of doors as we ENTER and LEAVE those rooms).
Let’s
suppose we start in a room with an odd number of
doors, then it is possible to visit all the 5 rooms
of the apartment if and ONLY if another room has
an odd number of doors  representing the
departure and the arrival points of the continuous
path  , and all the other rooms have an
even number of doors. In a few words, for this topological
puzzle to be solvable, there may NOT be more than
TWO rooms with an odd number of doors. Since the
puzzle has THREE rooms with an odd number of openings/doors,
it is mathematically impossible to complete a circuit
crossing.
Analogously,
a continuous line that enters and leaves one of the
rooms crosses two walls. Since the THREE contiguous
larger rooms each have an odd number of walls to
be crossed, it follows that an END of a line must
be inside each of them if all the 16 walls are crossed.
But a unicursal line has only TWO ends, this
contradiction makes the 5 Room House puzzle unsolvable.
However,
if we close a door or add an extra room to the puzzle
(see fig. a and b below), then
it becomes solvable. Now, you can easily draw one
continuous line that passes through every opening
exactly once... Try it! (the fiveroom variant on
the left (fig. a) is just a little harder to solve,
because you have to figure out where to start)
The
Five Room House is actually a classic example of
an impossible puzzle — one that bears no positive
solution. In this particular case, the solution
consists in finding that the problem has no solution!
(remember: puzzles always have one, several or no solutions;
see tips
to puzzle solving)
Graph
theory
The
insolubility of the 5 Room House problem can be proved
using a graph
theory approach, with each room being a vertex and
each wall being an edge of the graph (see
image opposite). In fact, this puzzle is similar to
the famous “seven
bridges of Königsberg” problem thanks
to which the eminent Swiss mathematician Leonhard
Euler laid the foundations of graph theory.
Euler wondered whether there was a way of traversing
each of the 7 bridges over the river Pregel at Königsberg (now
Kaliningrad) once and only once, starting and returning
at the same point in the town. He finally realized
that the problem had no solutions!

Tricks
to 'solve' the puzzle
As you experienced, this puzzle is impossible to solve
on paper... But ‘impossible’ puzzles sometimes
have outofthebox solutions, as the nonstandard
solution depicted below.
Another
neat outofthe box solution...
Everything down to this point has been in 2 dimensions,
either a diagram drawn on paper, or a five room apartment
on a flat surface. In order to draw a continuous path
that goes from one room to another without crossing
a line or going through a door twice, you have to reproduce
the 5 room house puzzle onto a surface that is not
topologically equivalent to a sheet of paper. The solid
that may help you is a torus, a kind of ringshaped
solid resembling a doughnut or a bagel. The puzzle
diagram should be reproduced so that the hole of the
torus is inside one of the 3 larger rooms, as shown
in the example below.
All
the Most Wanted Puzzle Solutions in a look!
