Renumbering levels


Often, when constructing a maze with students calling out the next number to use, the maze ends up with fewer than the planned number of levels. For example, suppose we are trying to build a 10-level maze and we get to

     0   9   4   7   8   1   2   3          10.
    ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___

Levels 5 and 6 are inaccessible. There are two things that can be done. One is to analyze the process to see where the 5 and 6 levels got cut off, and to change the construction. For example, here once the 8 is chosen there is no way to get to level 5. But a different choice can lead to a 10-level maze. Here one would have to choose 6 after 7, and that could lead to

     0   9   4   7   6   5   8   3   2   1  10.
    ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___

Another strategy is to decide to live with the 8-level maze you have constructed. Since levels 5 and 6 were not used, levels 7, 8, 9, 10 have to be renumbered as 5, 6, 7, 8. This yields

     0   7   4   5   6   1   2   3   8.
    ___ ___ ___ ___ ___ ___ ___ ___ ___


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Tony Phillips
Math Dept SUNY Stony Brook
tony@math.sunysb.edu
May 18, 1997