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Are you looking desperately for a puzzle solution? No panic, mail us your question and we'll try to solve your puzzle problem!


Puzzle archive
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External links
small square Walls and Lines puzzle on Wikipedia
small square 5 Room House puzzle explained on Maze.com
small square "Matematical Recreations" by Rouse Ball contains a rich section on 'Unicursal Problems'. The PDF version of the book is available here.
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Most Wanted Puzzle Solutions

 
The 5 Room House Puzzle

Solutions for your logic and mechanical puzzles

Walls and lines puzzleshoked face"Can anyone solve the following puzzle? It's simple as hell, but I can't figure it out... The idea is, you are supposed to draw this grid on a piece of paper, then draw one continuous line, starting anywhere, which crosses every grid bar once. I spent a long time on this problem, but still have no solution, please help me!" -- Robert

Category: Topological game / Unicursal and route problems.
Name: Five room puzzle, or Walls and lines puzzle.
Material: Pencil, piece of paper.
Aim of the game: Draw one continuous line that passes through every segment of the closed network exactly once.
Origin of the puzzle: Unknown. Martin Gardner described it in his book "Scientific American Book of Mathematical Puzzles and Diversions" (1957).
Editor's notice: As such, this puzzle CANNOT be solved.

 
SOLUTION
5 room puzzle variants
5 rooms puzzle
diagram with 16 openings/doors
5 room puzzle bis
3D rendering of the puzzle
tridimensional problem
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 five-room variant on the left (fig. a) is just a little harder to solve, because you have to figure out where to start)

solvable 5 room puzzles

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
graph of 5 room puzzleThe 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!

 

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brain and gearsTricks to 'solve' the puzzle
As you experienced, this puzzle is impossible to solve on paper... But ‘impossible’ puzzles sometimes have out-of-the-box solutions, as the non-standard solution depicted below.

cheating...

Another neat out-of-the 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 ring-shaped 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.

torus

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