## 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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