An Invisible Maze is a square room with a tiled floor, in which the tiles form a grid. You may walk only to adjacent tiles (no diagonal moves). There is a number on the wall for each row and column of tiles. An Invisible Maze can have any numbers on the walls provided that it has at least one True Path. A True Path will take you from the northwest corner to the southeast corner, and the number of tiles you touch in each row and column is equal to the corresponding number on the wall.

There is an NxN tiled Invisible maze that has at least two different True Paths. Minimize N and then, using that N, minimize the sum of all the numbers on the wall.

Important: Two paths are considered the same even if they touch the exact same tiles in a different order.

I suspect 7x7 is the minimum N, but the sum of 90 is probably not minimal.  The path is marked by x's.  The end of a row of x's would indicate going up or down a row (best I could do).  The *'s indicate unused spaces.  Note they are different in both solutions.

 

      7      7      6      6       6      7      6 

7    xxxxxxxxxxxxxxxxxxxxxxxxxxxx

7    xxxxxxxxxxxxxxxxxxxxxxxxxxxx

6    xxxxxx      *     xxxxxx     xxxxxx

7    xxxxxx      xxxxx       xxxxx       x

6    xxxxxx      xxxxx       *     xxxxxx

6    xxxxxx      xxxxx       xxxxx       *

6    xxxxxxxxxx      *       xxxxxxxxxx

 

      7      7      6      6       6      7      6 

7    x       xxxxx      xxxxxx      xxxxx

7    x       x      x      x       xxxxx      x

6    x       x      xxxxxx      *     xxxxxx

7    x       xxxxxxxxxxxxxx      xxxxxx

6    xxxxxx      *       xxxxx     xxxxxx

6    xxxxxx      xxxxx       xxxxx       *

6    xxxxxxxxxx      *       xxxxxxxxxx

 

I hope this is readable enough.

 

-Jer

Posted by Jer on 2004-11-01 17:55:53