## Constructing a new 10-level labyrinth.

In this phase of the activity, the class will work together to ``build'' (mathematically) a 10-level labyrinth. Since there are 162 different 10-level labyrinths, this might be a labyrinth no one has ever seen before. Building a 12-level labyrinth goes exactly the same way, but takes more time to prepare and to draw. If you are short of time or space you can easily adapt these instructions to make an 8, or even a 6-level labyrinth.

The teacher will coordinate and record this activity.

Preparation: Teacher draws 11 slots on the board, and writes "0" in the first and "10" in the last.

```
0                                       10.
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___

```
Teacher draws a number-line with the points 0,1,..., 10 labeled.
```

--+---+---+---+---+---+---+---+---+---+---+--
0   1   2   3   4   5   6   7   8   9   10

```
Teacher reminds the class of the three conditions:
• Starts with 0, ends with n, uses all numbers between 0 and n;
• Odds and evens alternate;
• No-crossing condition.

Producing the next number. "What do I put in this first slot?" Students will figure out that it has to be odd. Hold out for a fairly large one, like 7 or 9 (makes maze more interesting). When the number is decided on, write it in the slot and draw the corresponding segment in the appropriate place above the number-line. Go on to the next number, which will be even. When it is chosen, write it in its slot and draw the corresponding segment below the number-line. Suppose the first choice is 9, and the second choice is 4. then the number-line diagram will show

```
_____________________________________
--+---+---+---+---+---+---+---+---+---+---+--
0   1   2   3   4   5   6   7   8   9   10
_____________________
```
Continue until all the slots are filled. For example, if the sequence chosen is
` 0 9 4 7 6 5 8 1 2 3 10, `
then the number-line picture will be
```              ____
____________________________
____
_____________
_____________________________________
--+---+---+---+---+---+---+---+---+---+---+--
0   1   2   3   4   5   6   7   8   9   10
_____       _____________________
_____
_____________
______________________________
```

Getting stuck. Fairly often, there is no way to choose another number that is compatible with the rules. What has happened is that the class has found a smaller maze. If you want to keep it, you should renumber the entries to "count around" the unused numbers. Otherwise start over, or look at the number-line picture to see where a different choice would solve the problem. With a little experience, the teacher can see these problems coming and can "guide" the choices to avoid them.

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