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 10Teacher 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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November 10 1997, corrected April 9 2017 (previously: "Teacher writes '1' in the first [slot] ...").