Through Mazes to Mathematics

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## Exploring Labyrinths

This activity was carried out with a class of 4th-graders. It can be done in one 50-minute period.

Equipment needed:

• Each participant should have a sheet of paper (preferably printed with the "Cretan" nucleus on one side, and with the "walk-through" nucleus (see below) on the other. These two pages are available here: Cretan and walk-through.
• Each participant should have two different-colored pencils.
• Optional - images of the Cretan maze on greek coins, native american baskets, etc. These can be found in several different books, for example tony at math.stonybrook.edu
Matthews, Bord, Kern.
• Newsprint: enough to cover an 18x24 foot area.
• Wide masking tape - you will need at least 150 feet.
• Three or four fat-tip marker pens (for drawing on ground).
• Optional - music for the maze-walk.
1. Draw the Cretan maze (as shown here).
When I did it the kids all had pieces of paper with the "nucleus" already printed out. (on a 8.5x11 sheet the nucleus should be about 2x2 inches and placed 1.5 inches below the top of the page).
It may be useful to number the free ends from 1 to 8 on each side, starting at the bottom:
```

8   8   7
7.   |   |   |   .6
|   |   |
6 _____|   |   |_____ 5
|
5 _________|_________ 4
|
4 _____    |    _____ 3
|   |   |
3.   |   |   |   .2
|   |   |
2   1   1

The nucleus for the Cretan maze.
```
so then you can explain: "join 1 to 1, then join 2 to 2, ..., always going around the bottom and leaving space for the path."

2. Show them that the completed picture gives a path which winds back and forth and ends up at the center. Have them trace out the path with a different-colored pencil on their own copies.

3. Optional: show them pictures of occurences of the design they have just drawn on ancient coins, on a column in Pompei, on native american baskets, etc.

4. Tell them that we're going to draw a different maze, one we can walk through. They can have this nucleus on the back of the same page:

```
7
|   6
7   |   |   |   5
6   |   |   |   |   |   4
5 .   |   |   |   |   |   |   |   . 3
4   |   |   |   |   |   2
3   |   |   |   1
2   |
1

The nucleus for the "walk-through" maze.
```
Just as before, join up the free ends 1 to 1, 2 to 2, etc always around the bottom and leaving space for the path (i.e. keep the lines far enough apart from each other). Notice that there is an unlabeled end at the bottom and one at the top. These do not get joined to anything. You should end up with something that looks like this
```                              _______________________________
|                               |
|    _______________________    |
|   |                       |   |
_______________    |   |   |    _______________    |   |
|               |   |   |   |   |               |   |   |
|    _______    |   |   |   |   |    _______    |   |   |
|   |       |   |   |   |   |   |   |       |   |   |   |
|   |   .   |   |   |   |   |   |   |   .   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |       |   |   |   |   |   |   |
|   |   |   |   |   |   |_______|   |   |   |   |   |   |
|   |   |   |   |   |               |   |   |   |   |   |
|   |   |   |   |   |_______________|   |   |   |   |   |
|   |   |   |   |                       |   |   |   |   |
|   |   |   |   |_______________________|   |   |   |   |
|   |   |   |                               |   |   |   |
|   |   |   |_______________________________|   |   |   |
|   |   |                                       |   |   |
|   |   |_______________________________________|   |   |
|   |                                               |   |
|   |_______________________________________________|   |
|                                                       |
|_______________________________________________________|

```
(where I have picked out the nucleus in red) except the lines will not necessarily be straight. This maze has an entrance and an exit (it can be run either way).

5. Now we're going to draw this maze on the floor and walk through it. (The advantage of this one over the Cretan maze is that everyone does not end up bunched in the middle.) Let's calculate how big a space we need if the paths are going to be 18 inches wide. It is 14 paths across and 11 paths high, which means 21 feet across and 17.5 feet wide. (The kids can do this calculation.)
To draw the maze on the floor without destroying anything I roll out newsprint. Roll ends are available from your local newspaper printshop - they're usually happy to give them away. The kids roll out the paper and cut it while I put masking tape over the edges between strips. Good if everyone takes their shoes off for this. Then I draw the nucleus as above (leaving some space for the paths that go over the top: practice on a piece of paper) with 18 inches between the strips, and I draw the first (1 to 1) link on the bottom. The kids take turns drawing in the other parts of the path (I bring several thick markers). They just need some supervision to keep the path wide enough.

6. Finally we're ready for the perambulation. Make sure everyone has their shoes off; otherwise the paper will get torn. The kids line up at the entrance, each with his/her hands on the hips of the kid in front. Choose someone steady to lead the line. It is important to walk through slowly and methodically, enjoying the experience. When I did it with my daughter's 4th grade I had a friend along who played maze-walking music on the piano. Otherwise I've used Satie's "Gymnopedies" which is perfect.

7. For further exploration: take the walk-through nucleus and join the ends differently: join 1 to the next end on the right, and keep on joining the next two free ends around the bottom. What do you get?

Tony Phillips
Math Dept SUNY Stony Brook
tony at math.sunysb.edu
January 16 2001