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.

(In reply to re: 4 X 4 Probably Min by SteveH)

I like the no crossing idea better :-) 

But given that this solution is fair, it suggests a minimum sum for both walls for any N X N room of 2(2N+1). To see this, generalize the following example:

X X _ _  _
X X X X X
_ _ X _  X
_ _ _ _  X
_ _ _ _  X

X _ X _ _
X X X X X
_ X _ _  X
_ _ _ _  X
_ _ _ _  X

Edited on November 2, 2004, 12:13 am

Posted by owl on 2004-11-01 23:13:45