Background

Consider an $$\n\times n\$$ grid, where each cell is either empty (denoted by .) or a wall (denoted by #). We say that two cells are adjacent if they share an edge (a corner is not enough). A path is a sequence of distinct empty cells $$\s_0,s_1,\ldots,s_k\$$ such that cells $$\s_i\$$ and $$\s_{i+1}\$$ are adjacent for $$\0\le i.

We say that a grid is a maze if the following conditions hold.

1. There is at most one path between any two empty cells.
2. The following two $$\2\times 2\$$ patterns do not occur anywhere in the grid.
  #.             .#
.#             #.


A maze is not required to be connected (i.e. it is allowed to have two empty cells without any paths between them).

Here are a few examples of mazes.

.###                 #...                 ....
.###                 ###.                 #.##
.#..                 ..#.                 #.##
.###                 #...                 ....


Here are a few examples of grids which are not mazes.

....                 ##.#      Do not satisfy rule 1.
#.#.                 #..#
#...                 #..#
##.#                 #.##

##..                 ##.#      Do not satisfy rule 2.
..#.                 ##.#
#.#.                 .#.#
#...                 #..#


Challenge

Here is a $$\128\times 128\$$ maze. It has $$\991\$$ empty cells and $$\15393\$$ walls. Your task is to tear down as many walls as possible, making sure that the grid you obtain is still a maze.

More formally, let $$\M\$$ be the maze given above. A submission to this challenge consists of a $$\128\times 128\$$ maze $$\M'\$$ such that every empty cell in $$\M\$$ is also empty in $$\M'\$$.

Scoring

This is : the more walls you manage to tear down, the better.

Your score will be equal to the number of walls (#) in the maze you submit. Lower score is better.

Notes

• The maze $$\M\$$ given above doesn't have any special properties, to the best of my knowledge. It was generated as randomly as possible, fixing the average density of empty cells.
• Here is a python script that checks whether your submission is valid and tells you your score (just paste the maze in the "Input" section)
• Suggest an invalid case because of 2*2 .
– l4m2
Apr 21, 2021 at 20:38
• I'm pretty sure that this challenge is going to have many ties, since the optimum solution is any tree. Maybe it would be more interesting as code-golf. Apr 21, 2021 at 20:58
• @xigoi If you are correct, I'm going to be very sad :(. I won't change the scoring criterion now, but maybe in the future I'll post a new challenge. Apr 21, 2021 at 21:09
• @xigoi However, I believe things may be a bit more complicated than you think. For instance, consider these mazes. They are both maximal trees, but they have a different number of empty cells. Apr 21, 2021 at 22:36
• I think the score would be a little easier to understand and test if it were just the number of walls in the final maze, with lower being better.
– xnor
Apr 21, 2021 at 23:20

Score: 8066 8065 walls

I found an 8066 wall solution using dynamic programming over solutions that can be recursively decomposed by splitting on a row or column consisting entirely of floors or entirely of walls. The implementation of this apparently simple idea was rather more complicated than its description, since there are 17 different types of subproblems depending on the borders and the connectivity between them.

I then removed one wall by hand-editing seven cells in rows 119–122, columns 60–63.

This is an optimal solution. Proof:

Add a one-cell border using $$\516\$$ extra walls. Consider the planar graph whose vertices are the $$\w\$$ centers of walls, and whose edges are the $$\a\$$ wall-wall adjacencies. This graph is connected (because any component of walls not connected to the border would be encircled by a forbidden cycle of floors), and has $$\f \ge 2\$$ faces (at least an inside and outside). By Euler’s formula, $$\w - a + f = 2\$$, so $$\w - a \le 0\$$.

Now consider the different planar graph whose vertices are the $$\131^2\$$ corner points of walls, and whose edges are the $$\4w - a\$$ sides of walls. None of these $$\131^2\$$ vertices are missing (because an isolated one would again be encircled by a forbidden cycle of floors). This graph is again connected and has $$\f' \ge w + 2\$$ faces. By Euler’s formula, $$\131^2 - (4w - a) + f' = 2\$$, so $$\131^2 - 3w + a \le 0\$$.

Adding these two inequalities gives $$\131^2 - 2w \le 0\$$, so $$\w \ge \frac{131^2}{2} = 8064.5 + 516\$$.

Validate

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Score: 8066

This is the unedited solution from the dynamic programming algorithm.

Validate

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• Back in 2014 you claimed an incredible 17 character score on Glob of Regex Golf. Nobody has come close; somebody got 21 eventually; I've only been able to get 22 (the best published one is 23). Could you please share your amazing solution now? I think it's been long enough :) Or give a hint as to what schema was used. It would be sad for your solution to disappear into the mists of time. (You've ignored my emails, so I'm reduced to trying this now.) Apr 22, 2021 at 17:32

Score: 8127 (validation)

-11 thanks to Anders Kaseorg

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• Feed this back into your script again—you can knock down another 11 walls. Apr 22, 2021 at 1:01
• @AndersKaseorg Nice, never thought to try that. Apr 22, 2021 at 1:08