Possibly silly question... I played more than 100 puzzles and casually browsed many more, but I couldn't find one with a symmetry axis. Is it just because they are extremely uncommon, or did you exclude them for some reason?
There are a couple examples linked in another comment[1], so they do exist. The author of the game has stated[2] they excluded about 1/4 of all possible configurations to avoid including puzzles that required too much trial and error. Perhaps symmetry leads to more ambiguity than asymmetry, and therefore more than ~1/4 of symmetrical configurations were excluded?
Your comment made me curious about how often symmetry occurs in the full set of all possible 5x5 configurations. I took a shot at calculating this as an exercise, but I am a bit rusty when it comes to combinatorics...
First consider mirror symmetry via the center column. There are 2^5 configurations of the center column, and for each of those, there are 2^10 configurations of the left two columns. Since we are mirroring the right two columns from the left two, the number of configurations exhibiting mirror symmetry via the center column is 2^5 * 2^10 = 2^15. Rotating these 90 degrees gives us mirror symmetry via the center row, which is another 2^15 configurations. Mirror symmetry via the corner-to-corner axes, which also have 5 squares, is another pair of 2^15 configurations. So now we're at 2^17 configurations for mirror symmetry for the four axes.
Radial symmetry is slightly harder to describe, but it involves similar partitioning. You can partition the 5x5 grid into two 12-square subsets excluding the center square:
x x x x x
x x x x x
x x . o o
o o o o o
o o o o o
For any given configuration of the x-subset, you flip and reverse that to get the configuration of the o-subset. There are 2^12 possible configurations of the x-subset. Since there are two possible values of the center square, that gives us 2^13 configurations of two-subset radial symmetry. I believe rotating 90 or 180 degrees simply produces another configuration that has already been accounted for.
There is also four-subset radial symmetry:
a a a B B
a a B B B
D a . c B
D D D c c
D D c c c
however, I think these would all be special cases of two-subset radial symmetry. If I pick a random configuration for the a-partition and apply it to the other subsets, it matches a configuration that would appear in the two-subset group:
a a a B B X X o B B X X o o X
a a B B B o X B B B o X X X X
D a . c B => D X . c B => o X . X o
D D D c c D D D c c X X X X o
D D c c c D D c c c X o o X X
So between mirror symmetry and radial symmetry we have: 2^17 + 2^13 = 139,264. There are a total of 2^25 = 33,554,432 configurations irrespective of symmetry, so that's 17/4096 or roughly 0.415% that are symmetric...a bit more than 1/256.
EDIT: And by some hilarious bit of fate, I just went to go knock out a few puzzles to reset my brain, and the first one I completed[3] exhibits mirror symmetry in one of the diagonal axes. I'm pretty sure this is the first one I've hit in over 1000 solves.
Your comment made me curious about how often symmetry occurs in the full set of all possible 5x5 configurations. I took a shot at calculating this as an exercise, but I am a bit rusty when it comes to combinatorics...
First consider mirror symmetry via the center column. There are 2^5 configurations of the center column, and for each of those, there are 2^10 configurations of the left two columns. Since we are mirroring the right two columns from the left two, the number of configurations exhibiting mirror symmetry via the center column is 2^5 * 2^10 = 2^15. Rotating these 90 degrees gives us mirror symmetry via the center row, which is another 2^15 configurations. Mirror symmetry via the corner-to-corner axes, which also have 5 squares, is another pair of 2^15 configurations. So now we're at 2^17 configurations for mirror symmetry for the four axes.
Radial symmetry is slightly harder to describe, but it involves similar partitioning. You can partition the 5x5 grid into two 12-square subsets excluding the center square:
For any given configuration of the x-subset, you flip and reverse that to get the configuration of the o-subset. There are 2^12 possible configurations of the x-subset. Since there are two possible values of the center square, that gives us 2^13 configurations of two-subset radial symmetry. I believe rotating 90 or 180 degrees simply produces another configuration that has already been accounted for.There is also four-subset radial symmetry:
however, I think these would all be special cases of two-subset radial symmetry. If I pick a random configuration for the a-partition and apply it to the other subsets, it matches a configuration that would appear in the two-subset group: So between mirror symmetry and radial symmetry we have: 2^17 + 2^13 = 139,264. There are a total of 2^25 = 33,554,432 configurations irrespective of symmetry, so that's 17/4096 or roughly 0.415% that are symmetric...a bit more than 1/256.EDIT: And by some hilarious bit of fate, I just went to go knock out a few puzzles to reset my brain, and the first one I completed[3] exhibits mirror symmetry in one of the diagonal axes. I'm pretty sure this is the first one I've hit in over 1000 solves.
[1]: https://news.ycombinator.com/item?id=44148396
[2]: https://news.ycombinator.com/item?id=44141047
[3]: https://pixelogic.app/every-5x5-nonogram#3328511
EDIT: formatting; add link to symmetric puzzle