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the Five-Room puzzle

Many of us have been stymied by The Five Room House Puzzle Can you draw one continuous line that passes through every opening exactly once. Real Life Instructions: Can you walk through every door once? (close each door after you go through it)

In essence, this diagram shows a six-room house. The outside of the house counts topologically as the sixth room.

If this were a real house, your task would be to start inside any room (or inside the outside room), and walk through every door exactly once without using any door a second time. (If you want an easy way to keep track, start by opening every door, then when you are ready to solve the puzzle, start walking through and around the house, closing each door as you pass through it, so you can't use it again.)

On paper, your task is to start inside one "room" and draw one continous line that passes through every opening without crossing any lines (and without going through any opening twice). You may, of course, end in any room of your choice, not necessarily where you started.

This puzzle is impossible
it cannot be solved (on paper)
(but further down, I'll help you cheat)

Before I continue, let me start by telling you that this puzzle can NOT be solved. It is an impossible puzzle, and we can use mathematics to demonstrate why it cannot be solved, unless you cheat (and yes, further down, I will help you find a way to cheat on this puzzle that even your teacher will accept).

Two variations of this puzzle
that can easily be solved

We could, of course, design a house that would allow you to solve the puzzle with no difficulty at all. Let me show you two houses that can actually be solved:

and
six-room solvable variant            five-room asymmetrical variant

The six-room puzzle on the left is very easy to solve. Start anywhere, go through every door once, and you'll end up back in the room where you started.

The five-room non-symmetrical puzzle on the right is just a little harder to solve, because you have to figure out where to start. If you start in the wrong room, you won't be able to go through all the doors. But if you start in the right room, you'll be able to go through all the doors, but you'll end up in a different room from where you started.

BEFORE YOU CONTINUE, please solve both of these puzzles. (I promise, both of these variants can be solved, and if you have trouble solving one of them, I'll tell you more about what you need to know further below.)

Requirements for solveability

In order for this type of puzzle to be solvable, we need to either:

1. Start in a room with an even number of doors, and end up back in that very same room, or
2. Start in a room with an odd number of doors, then end up in a different room that also has an odd number of doors.

But, whether you start and end in the same room, or you start in one room and end in another room, every other room in the house must have an even number of doors. Stop for a moment, and try to answer, for yourself, why this must be.

(pause for humming while you're thinking)

I bet you figured out that every other room must have an even number of doors, because you'll use an even number of doors as you enter and leave those rooms. If you think back to the real-life example at the top of the page, as you enter the room, you close a door behind you, and as you leave the room, you close another door. Entering and exiting a room involves closing two doors. Or, drawing a line through a room on the chart involves making two openings unusable. There is no way to only use one door as you pass through a room, nor is there a way to use three doors. Since you must use (or consume) two doors as you pass through a room, there must be an even number of doors.

So, if you want to start in a room with an even number of doors, you must have a house in which EVERY room has an EVEN number of doors. The six-room variant puzzle is such a puzzle. To solve it, pick any room as your starting point, go through all the doors, and you'll end up back in the room where you started.

If you want to be able to start in a room with an odd number of doors, you must have a house in which ONLY two rooms have an odd number of doors, and all the other rooms have an even number of doors. The non-symmetrical 5-room puzzle variant is such a puzzle. To solve that puzzle, you must start either outside the house (the outside has nine doors), or inside the room with five doors. After you go through all the doors, you'll end up in the other "odd" room.

To summarize this section, for one of these puzzles to be solvable, there may NOT be more than two rooms that have an odd number of doors. (I challenge you to create a house which only has one "room" with an odd number of doors ... remember, the outside of the house also counts as a room.)

Why is the official puzzle
impossible to solve?

If we count the doors in that lead into each room in our original puzzle, we will find out that there are four odd numbers. Let's count the rooms right now. For my "count", I am going to draw lines that show where each door leads. For the outside of the puzzle, I'm going to pretend that we have to walk to a specific point outside the puzzle and put the count for that room in that place.

As you can see, we have three rooms with five doors, and one room with nine doors. Based on what we discussed above, this puzzle cannot be solved because there are more than two odd numbers. Even if we start in one of the odd numbered rooms, we will eventually get to the point when we are trapped inside one of the other odd number rooms with no escape possible.

Let's say that we start outside the puzzle. As we go inside the house, the outside number changes to 8 (to show that there are now only 8 doors available to be used now), and the number inside the room we enter reduces by 1, and when we leave that room, it again reduces by 1. Each time we enter-and-leave one of the other three odd-numbered rooms, those numbers go down by 2, and eventually we will have three 1's inside the diagram. That means we must avoid those three rooms until we have done everything else because if we enter a room with a 1, that 1 becomes zero, and we are trapped, and unable to get to either of the other odd rooms.

For a more detailed explanation of Matthew Euler and his method of proving that this puzzle is impossible to solve, see my other "Five Room Puzzle" page, click here, or use the link at the bottom of this page after you've read the rest of this page.

Cheating in two dimensions

Cheating on paper: If we close one of the doors between two odd-numbered rooms before we start the puzzle, we can cheat and solve the puzzle. However, every math teacher I know will not accept this method of cheating, because, for all practical purposes, it is the same as not going through one door. You could, of course, make up a big story about how that was the only acceptable place to hang your new high-definition wide-screen television. Here is a picture of my five room house with my new TV and entertainment system (in red). The green area is the access panel that lets me rewire the back of the amplifier unit.

Now, because you're not allowed to walk through the TV (it's floor to ceiling, so you can't even climb over it), the top left room only has 4 doors, and the middle bottom room only has 4 doors. Now, the only odd rooms are the upper right room (5 doors) and the outside (9 doors). If you start outside, and end in the bottom right room (or vice-versa), you can solve this puzzle by cheating (by putting up a nice new television and entertainment system that won't otherwise fit in the house. But as I said, your teacher won't accept this method of cheating unless you are very clever in explaining why you needed to remodel the house.

Some people will try to cheat by going "vertically through a door" like in the enlargement of one door that I've displayed to the right, but teachers also call this cheating, and I can't suggest a good reason for trying to justify this to a teacher. It goes "through" the area in line with the wall, but it does not go through the door.

BUT WAIT A MINUTE ... there is a solution to this puzzle after all.

Cheating in Three Dimensions

Can you solve this puzzle? Draw this diagram onto some kind of surface (after all, you don't want to mess up your screen), then draw one continuous line that passes through every opening without crossing any of the lines. It can be done. I've done it. But you have got to be creative. That's doubly-emphasized underlined, bolded and italicized creative. Don't try to do it on a piece of paper. It can't be done.

As I've hinted above, there are methods of cheating that you can probably convince a teacher is a valid method of solving the puzzle, but they require changing out method of thinking. Everything down to this point has been in two dimensions, either a diagram drawn on paper, or a house on a flat surface.

In order to really cheat successfully, we need to think in three dimensions. Look around your house for objects that might help you. Or you might find something down at the local deli that will help you.

Basically, you need to think topologically, and draw the puzzle onto a surface that is not topologically equivalent to a sheet of paper. You need something with three dimensions, but not a ball. It has to be slightly more involved than a simple solid.

There are two methods that you can use to cheat on this puzzle, one by thinking about actually being able to walk through the puzzle (think real life, maybe even your home, depending on how it's arranged, but not simply by opening a window and jumping outside), and the other by thinking about what kind of topological objects that you can draw the puzzle onto, objects that will allow you to solve the puzzle, because they will allow you to draw a continuous line that goes from one room to another without crossing a line or going through a door.

For a more detailed example of how to cheat in both of these methods, click here to get to my Five Rooms page that also gives a better example of Euler and his explanation about why this puzzle cannot be solved. It also includes a similar but simpler puzzle called the Bridges of Konigsberg, and tells you exactly how to cheat with both puzzles.

If you enjoy puzzles, please visit <www.MAZES.com>. If you'd like to thank me for the time that went into creating this page full of information (many days of effort), please take advantage of our links. For example, if you are going to purchase some books from Amazon.com, click our links first. We will prepare other pages of information about ways that you can help us. Check back and see what else you can do (and feel free to make donations).

John
(webmaster (at) mazes.com)
www.MAZES.com

P.S. People who enjoy turning phone numbers into words might be interested in knowing that our area code spells GRY, certainly an appropriate area code for the webmaster of www.MAZES.com, a puzzle page. Unfortunately, I couldn't get MAZEMAN as a phone number (because our phones don't have Z's on them), but I did get BYTEMAN. So, I guess that I can call myself "1-GRY-BYTEMAN".

The GRY collection at www.MAZES.com:
Our webmaster's thorough explanation One Hundred Words that end in -gry: Another GRY expert and his explanation A third expert's GRY point of view	www.MAZES.com/gry.htm (this page) www.MAZES.com/gry-2.htm www.MAZES.com/gry-3.htm http://www.word-detective.com/gry.html