Map Puzzle - Can you create a map with five regions all touching each other

Five-Region Map Puzzle

Can you create a map with five states that all touch each other?
one possible answer, with cheating

The challenge: Can you create a map that contains five regions or areas in such a way that every region touches every other region along a common shared border (along a line, not at a point). The areas do not need to be equal.

Feel free to stop here, and try to figure it out for yourself. When you are ready to continue, keep reading.

First, draw a region of any shape to be our first "state" but keep it reasonably small. I'll do it by typing, and use 1's to show the area. For example.

Second, draw a second region that touches the first. Let's let it go slightly above and below the first.

Now, we need a third area that touches, shares borders, with both one and two. Let's put it above 1 and slightly around 2.

Now, we get to the tricky part. If we start our fourth state so that it touches both 1 and 3 where they meet, and draw it going down to 2, it will completely surround 1, and nothing will be able to get inside.

      333333333
      333333333
   444333333333
   444333222222
   444333222222
   444111111222
   444111111222
   444111111222
   444111111222
   444111111222
   444444222222
   444444222222

And if you tried to start where 2 and 3 meet, and go up and around to touch the 1, then 3 will be completely surrounded.

   444444444444444
   444444444444444
   444444444444444
   444333333333444
   444333333333444
   444333333333444
   444333222222444
   444333222222444
   444111111222444
   444111111222
   444111111222
      111111222
      111111222
         222222
         222222

The same is true if you start anywhere else. The fourth state prevents a fifth state from touching ONE of the first three. From what I can see, there is no method that we can use to draw five regions on a flat piece of paper. If you can find one, remember to e-mail me.

But I'm not saying that there is not an answer to this challenge. There is an answer. Let's erase the 4's from the above diagram and look again.

Okay, let me give you a hint.

We need a trick.

Take a break from the computer and walk around your house and look for something that you can draw this diagram on that would let you solve the puzzle by cheating.

Go ahead, do that, I'll wait for you.

You say you didn't see something? I'm sure you've got something there. Why don't you look in your swimming toy collection? Or you might go down to the neighborhood bakery and see if you see something useful there. Or look in the dishwasher or up on the shelves.

Give up?

Have you ever heard of a TORUS? A torus is like a donut or a bagel. A coffee cup is an example of a torus as long as it has a handle that touches the cup in two different places. Topologically, a coffee cup is identical to a doughnut because you could massage a soft-clay coffee cup and turn it into a doughnut without tearing aprt or putting any parts together..

I'm sure that Lazarus Long, one of the heroes in Heinlein's Future History series, including some of my favorite books, including The Cat Who Walked Through Walls, has visited planets that were shaped like toruses, so you might try to invent such a planet for part of your demonstration. Or, create a planet shaped like a coffee cup. Anyway, let's get on this spaceship and travel to a planet that looks like a donut, bagel, or coffee cup.

While I get out the ship, I want you to go to the bakery and get the smoothest stale bagel you can find, one that you can draw on.

Ok, I see your back. Draw this diagram on one side of your bagel, so that you can see bagel all the way around the diagram.

    ________________________
   /
  /    333333333
 |     333333333               I'll let you 
 /     333333333      ______   imagine what
|      333222222     /         the other side
|      333222222    /          of the bagel 
|      111111222   |           looks like
|      111111222   |
|      111111222   \
 \     111111222    \_______
 |     111111222
  \       222222
  |       222222
   \
    \_______________________

Now, let's draw a band of 4's that starts on the left side, touching both the 3's and 1's and goes to the left, all the way around THAT PART of the bagel.

    ________________________
   /
  /    333333333
 |     333333333              
 /44444333333333      ______  
|444444333222222     /        
|444444333222222    /         
|444444111111222   |          
|444444111111222   |
|444444111111222   \
 \     111111222    \_______
 |     111111222
  \       222222
  |       222222
   \
    \_______________________

And keep going underneath the bagel, where I can't see right now,

(imagining you doing the work)

And come up on the other side, making it touch the 2's.

    ________________________
   /
  /    333333333
 |     333333333              
 /44444333333333      ______  
|44444433322222244444/        
|4444443332222224444/
|444444111111222444|          
|444444111111222444|
|444444111111222444\
 \     1111112224444\_______
 |     111111222
  \       222222
  |       222222
   \
    \_______________________

Now, can you figure out where to put a fifth country or region so that it touches EVERY OTHER region?

Stop and think about it. You'll kick yourself if you give up too quickly.

Scroll down for the answer

S
c
r
o
l
l

d
o
w
n

f
o
r

a
n
s
w
e
r

You've already got it, everything that you haven't drawn on the bagel touches everything else. You have your answer. Remember, the fours go completely around the bagel.

    ____________________________
   /5555555555555555555555555555\
  /555553333333335555555555555555\
 |55555333333333555555555555555555\
 /444443333333335555555555555555555\
|44444433322222244444/      \5555555\
|4444443332222224444/        \5555555\
|444444111111222444|          |5555555|
|444444111111222444|          |5555555|
|444444111111222444\         /55555555/
 \555551111112224444\_______/55555555/
 |5555511111122255555555555555555555/
  \55555552222225555555555555555555/
  |5555555222222555555555555555555/
   \55555555555555555555555555555/
    \555555555555555555555555555/

Here's another view of the bagel, aligned differently.

               5555555555555555555555555555
           555555555555555555555555555555555555
        555555555555555555555555555555555555555555
      5555555555555555555555555555555555555555555555
     5555555555555555555555    5555555555555555555555
    555555555555555555555         55555555555555555555
    55555555555555555555            555555555555555555
...444444333333333444444...          555555555555555555
...444444333222222444444...          555555555555555555
...444444333222222444444...          555555555555555555
...444444111111222444444...          555555555555555555
...444444111111222444444...          555555555555555555
...444444111111222444444...          555555555555555555
    55555111111222555555             55555555555555555
    555551111112225555555           555555555555555555
     5555555222222555555555       5555555555555555555
      5555552222225555555555555555555555555555555555
       55555555555555555555555555555555555555555555
         5555555555555555555555555555555555555555
            5555555555555555555555555555555555
                55555555555555555555555555

This should help you get extra credit. Remember, if you're going to eat your math homework later, use edible ink, but if you're not going to eat it, you can probably use five different colors of magic markers.

If you don't have any bagels, see if your mom will let you have one of her old coffee cups that still has a handle that touches the cup both places. A broken handle would ruin this project.

P.S. I wrote this web page as a volunteer (almost three hours after teaching at TutorNet.com, and more hours later, I'm sure. If you appreciate all the work that goes into presenting these puzzles, writing them up, and helping you solve them, then please visit <www.MAZES.com> and support it by visiting any advertisers you see there. And if you like personalized puzzles, look for your name at <www. GodLovesEveryone.org> and look for ways to earn free puzzles. Some of our advertisers purchase personalized puzzles for people who make purchases. Look for other free offers at <www.MAZES.com/FreeMaze>.

Five-Region Map Puzzle

Can you create a map with five states that all touch each other?one possible answer, with cheating

Can you create a map with five states that all touch each other?
one possible answer, with cheating