## Numbering the levels in a labyrinth.

In this phase of the activity students will discover the
*level sequence* of the Cretan maze, and will analyze
the relative positions of the numbers of the numbers in the sequence.
**First** the teacher leads the class in putting numbers
on the levels of the Cretan maze they have drawn. Put a 0
just under the maze, 1 in the middle of the lowest level,
2 in the middle of the one just above it, ... , and finally
an 8 in the center.

**Now** the students start at the outside (0) and go
through the maze writing down the numbers as they meet them.
Each student should do this privately, then they should
compare results. They all should have 0 3 2 1 4 7 6 5 8.

**Next:**``Do you notice anything special about the
way these numbers are ordered?'' Chances are, some students
will pick up on the fact that *odds and evens alternate*.

There is an **additional condition** that a sequence
starting with 0, going through the first n numbers
in some order, and ending with n, must satisfy in order
to give a maze: the
*no-crossing condition*.

Consider the pairs of consecutive
numbers in the level sequence that begin with an even number;
in the Cretan case these are (0,3) (2,1) (4,7) and (6.5);
these correspond to the vertical paths on
the right side of the maze. Draw the corresponding
segments over a number-line: [0,3], [2,1], etc. and
show the class that
*Whenever two of these segments overlap, one
is nested inside the other.* Show the class how
this condition guarantees that the path does not
cross itself when it changes levels on the right side
of the maze.

Now look at the pairs of consecutive numbers beginnig with
an odd number; in this case (3,2) (1,4) (7,6) and (5,8);
these correspond to vertical paths on the left side.
Draw the corresponding segments, this time below the
number-line, and show the class that they also
satisfy *Whenever two of these segments overlap, one
is nested inside the other.*

Any sequence that satisfies these conditions:

- Starts with 0, ends with n, uses all numbers between 0 and n;
- Odds and evens alternate;
- No-crossing condition

will give a labyrinth.

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*Tony Phillips*

March 14 1997