# 17x17: Some Attempts at Doubly-Symmetrical Rotations

Note: this is part of a series of posts is related to the " 17x17 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 a previous post
I presented rotations of some single-color rectangle-free grids which were symmetrical along a diagonal axis. I
also noticed that, of the single-colorings of optimal size which I could
generate, all with an odd number side-length could be made symmetrical along
*both* diagonals (the evens could not):

```
TODO: insert google doc iframe
```

Perhaps *all* odd-sided optimal single-colorings are doubly-symmetrical in one
of their rotations! That would be a cool thing to learn, and it would also
mean that if we wanted to generate an optimal coloring, then our search space
would be roughly 1/4 of the grid.

So I thought I would see if I could rotate the known 74-color grid into a doubly-symmetrical arrangement… and I have to admit defeat.:

```
TODO: insert google doc iframe
```

The orange squares mark discrepancies I couldn’t resolve. It could be that it
is true that all optimal (having the greatest number of colored cells possible
for the grid size) single colorings *can* be made doubly symmetrical and that
the 74 colors in 17x17 grid is less than optimal. It could also mean I suck at
moving rows and columns around in google spreadsheet. Either way, I’m done
with this line of investigation for now.

Next I would like to jump into actually trying to generate 4-colorings of 17x17, using some informed search algorithms and ladder climbing techniques.