Note: this is part of a series of posts is related to the “17×17 Challenge” posted by Bill Gasarch. The goal is to color cells of a 17 by 17 grid, using only four colors, such that no rectangle is formed from four cells of the same color.

In my last post, I gave a symmetrical rotation of the known 74-cell single-coloring. I want to use a slight variation of that grid to show why I think treating the grids as symmetrical will help us in solving this problem.

Here is a new spreadsheet, with several panes you can click through on the bottom. Panes ONE through FIVE represent the first few rows of a single-coloring in which we avoid making any real choices, thanks to a few assumptions we make (I’ll come back to that).

Orange represents the row we’re coloring, gray cells are rectangle forming cells (in which marks are not allowed), and blue cells represent what I consider to be the real search-space:

NOTE: We’re simply using another automorphism of the original 74-coloring, i.e. a series of rotations that preserve all the significant attributes of the graph.

Making Coloring Decisions

We can choose, arbitrarily, to start with a 5-cell row starting on the cell in the upper left. This is our first decision: if it were that in an optimal single-coloring of 17×17 no row with five colored cells shared a cell with any column with five colored cells, then we would have made a poor decision.

But given that in the 74-coloring that we have, all 5-colored-cell rows share a cell with a 5-colored column, we can probably assume that at least one will in our optimal single-coloring. So our first row (in pane 1) is a non-decision.

We continue with the second row by dumping 3 cells as soon as we can (i.e. as far to the left as possible. The only decision made here is in whether to make row two a row of four or five. But as we can see very quickly (and as a shallow search algorithm would see very quickly), making row 2 a five-colored-cell row would force us into forming a 3-cell row out of row five.

So in row 2 we have an easy decision (seeing that we would soon regret creating a 5-cell row) based on our assumption that a good singly-colored grid will have its cells spread evenly over rows & columns.

And we have the non-decision of the column placement of those three cells: because identical columns (in this case empty columns) can be freely swapped without changing anything, there is no need to try every combination of column placement for the row. We put off our decision making for later.

We continue on in the same way.

Search Ideas

I’m coding up a program to try to generate good grids based on some heuristics. I think that a minimax type solution might be good: in which the coloring of a cell is assigned a point-value based on how many future cells it makes unusable. I could then keep track of which previous cells were complicit in forming potential rectangles and back-track at apparently-bad decisions: i.e. hill climbing.

I may try to formulate the algorithm like a game and see if I can use the game-tree module by Colin Adams.

Conclusion

It seems to me that the key to the problem of a single-coloring is shrinking the search-space by eliminating false-choices. Treating the graph as it’s symmetrical (by our definition of symmetrical) automorphism is one way: suddenly the search space becomes the shaded area in the graph above.

Thankfully, my next post will be haskell-related and have nothing to do with Grids.

Single-Color Subsets

I’ve started by trying to find an algorithm to find optimal single-color subsets, i.e. a way of coloring a grid with one color such that no rectangle is formed by the colored cells, and we have the greatest number of cells colored as possible. I started thinking of coloring cells one at a time, following a path based on the notion of an “extended rectangle” where the fourth side was longer or shorter than the first, to avoid forming rectangles.

I also noticed that we could traverse all the cells in the largest known 17×17 subset by following a simple spiraling algorithm. It didn’t seem to matter which cell or direction we started in either:

go until you hit a colored cell, then rotate 90 degrees

Optimal Grids

I generated some optimally-colored grids to see if they all followed this property of being traversable via this “rotation” algorithm above. It turns out that some directions and starting points loop before they touch every colored cell. I suspect that will also be the case for the 74-color grid above.

Here are the grids I was able to generate via brute force:

Notice that, as you might guess, colored cells are spread out as evenly as possible among rows and columns (i.e. no two rows differ by more than one in their number of colored cells).

…and Symmetry and Rotations

Also notice the diagonal symmetry that all of the grids exhibit. The 5×5 grid was not symmetrical when it came out of my brute force algorithm, but I was happy to see that I could turn it into the above symmetrical version with a few rotations (a fun puzzle by the way). The symmetry reminds me of creating/solving sudoku and I wonder if this is a similar constraint-solving problem.

It would be interesting to see if the 17×17 grid was symmetrical in one of its possible rotations. I would like to look into that as well as code up a symmetrical variation of the algorithm I described in my last post. If all optimal subsets were symmetrical, that could make the search-space considerably smaller.

I also wonder whether all subsets of the same size are not simply rotations of the others. This seems to be the case for the smaller optimal subsets in the image above.

Four-Coloring the 17×17 Grid

The author suggests that a 17×17 grid is an edge case, meaning that it may or may not be possible to four-color. A single-coloring of at least 73 must be possible in order for a four-coloring to be possible: 73 + 72 + 72 + 72 = 289 = 17 x 17. The author of the challenge has found one of size 74.

Given the difficulty of finding a single-coloring (74 colors seems to be the largest known), we can perhaps assume that each of the four single-color subsets will be evenly distributed with 4 or 5 cells per row/column.

Thus every row of a successful four coloring will have 4 + 4 + 4 + 5 = 17 in each of the four colors respectively.

There are 2380 possible unique 17-length rows with four colored cells in the row. There are 6188 with 5 colored. Here’s the code I used for that fun fact:

-- finding promising rows:
allRowsSize sz n
    | sz == n = [ replicate n  True ]
    | n  == 0 = [ replicate sz False ]
    | otherwise =
       [ True:as  | as <- allRowsSize (sz-1) (n-1) ] 
        ++ [ False:as | as <- allRowsSize (sz-1) n ]

Going Forward

If the couple ideas I still have for an optimal single-coloring don’t work out, it may be interesting to try to “deconstruct” the known 74-coloring, treating it like the algorithm I developed in my last post, but in reverse, noticing the “choices” made.

After giving up on a deterministic single-coloring algorithm it might be interesting to investigate some kind of ladder climbing algorithm that looks at trying to consolidate rectangle-forming squares.

Finally, an approach that might have been the easiest all along, would be to take the author’s 74-coloring above and try to generate 4 different permutations of sizes 73 and 72 that could fit together. We don’t learn too much from that though.

I’m really interested to know your thoughts on the problem as well. So leave a comment if you like!