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The Mazing (Posted on 2003-10-02) Difficulty: 3 of 5
A maze is to be constructed on an X by Y grid of squares by creating "walls" between them.

The process starts with an enclosed rectangle X squares wide and Y squares tall, with walls on the outside edges of the outermost squares. Here's an example of a (6 x 2) rectangle:

       +--+--+--+--+--+--+
       |                 |
       +  +  +  +  +  +  +
       |                 |
       +--+--+--+--+--+--+
We then build additional walls in the following manner:
  • Pick an existing wall (A) at random.
  • Pick (also at random) a potential non-existent wall (W) that will be adjacent to (A). (W) must be a valid wall - it must be inside the maze.
  • Ensure that (W) will not connect wall (A) to any other wall - that is that there is no existing wall (B) that is is distinct from (A) and non-adjacent to (A) but that would be adjacent to (W).
  • Place wall (W) only if the above condition is satisfied.

    By following in these steps, what is the theoretical maximum number of walls that can be placed? Is there a situation where walls can be placed in such a way that achieving this maximum will no longer be possible?

  •   Submitted by levik    
    Rating: 3.3333 (3 votes)
    Solution: (Hide)
    The maximum is (X-1) × (Y-1).

    This means only one wall can be added to a 2×2 maze, 2 walls in a 3×2, 4 in 3×3, etc.

    This is because the number of walls that can be added is exactly equal to the number of 'internal' points to the rectangle, as Charlie explains here.

    Charlie also gave a program in BASIC that runs this algorithm, randomly picking walls and placing them to fill in a maze; that can be found here.

    Comments: ( You must be logged in to post comments.)
      Subject Author Date
    Going in CirclesBrian Smith2003-10-03 09:29:32
    why didn't my <pre> work?Charlie2003-10-02 22:25:40
    SolutionsolutionCharlie2003-10-02 22:16:07
    Hints/TipsFor startersCharlie2003-10-02 15:45:16
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