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 am using what I think nikki is getting at with  the numbers and Xs of the previous comment. N=1 cannot give 2 different True Paths. For N=2 it looks to me like

     2     2

2   X     X

2   X     X

works. Removing either the NE or the SW X will result in just one such path, and of course the NW and the SE X must each be there, so I don't think that a 1 can be on the wall. This looks like the minimum-minimum to me.

Since I am not sure I really understand what is wanted here, I am going to label this "a try," rather than a "solution."

Posted by Richard on 2004-11-01 22:41:59