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 Clarification questions by nikki)

"1) Are you allowed to touch one tile twice? For example, is the following path legal?"

No, you can only touch each tile once.  I wasn't aware of this ambiguity, and I apologize.

"2) So what you’re saying about the numbers on the walls, is this:

  4 3 5 1 3
3 X X X _ _
1 _ _ X _ _
5 X X X X X
3 X _ X _ X
4 X X X _ X

is that correct?"

Yes, the number of touched tiles in each row/column is equal to the number on the wall.

"3) So the question of the problem is that you have your NxN maze (no X’s exist yet) and the numbers on the walls indicating how many X’s there will be in that row/column, and you should be able to find two different arrangements of Xs, both of which are legal paths."

Exactly.

"4) The "Important" note means that if somewhere in my maze I have a 3x3 block of X’s, it doesn’t matter that I could go down-down-right-up-up-right-down-down as well as right-right-down-left-left-down-right-right, correct?"

Yes, though a 3x3 block of Xs is not the only thing that the important note keeps from counting as two paths.

Posted by Tristan on 2004-11-02 01:38:55