Puzzle #384 - A Learning Experience (Every Second Straight)

I believe that I've mentioned that I'm making 1 puzzle for every genre that PuzzleDuel has, to submit in bulk and provide quality puzzles for everyone. Well, I recently got to Every Second Straight, a loop genre that I only knew from the site. To make matters worse, I solved all prior examples by intuition and tweaking, and had no idea where to begin logically. I suspect many solvers of my puzzle were in the same boat, though I promise there's a clean path through this puzzle with the right knowledge.

puzz.link interface      penpa interface      PuzzleDuel interface

Solving tips after the break. And by tips, I mostly mean revealing the biggest secret this genre holds.

Alright, so the most important step in this puzzle is, naturally, one that requires parity. Select any given straight cell, and arbitrarily choose an orientation (left/right or up/down) and direction for the loop. Directing the loop isn't necessary, but it helps with thinking about the overall structure of the loop. If you color the grid like a checkerboard, the next cell will either be straight (uncircled, other color) or a turn (changing the orientation). Continuing will show that for a straight segment on any given checkerboard color, its orientation must be constant. The key insight is that the only way to change the parity of the orientations is to travel straight (as a turn changes both), but every other straight is unmarked. This actually goes a step further and allows you to say that if a cell does not contain a circle, then it can not go straight in the other orientation. For this puzzle, it can be used to say that R1C9 can not be a straight, because if it were, by parity it would have to be a circle.

That structural knowledge combined with a few short lookaheads utterly destroys this puzzle, and honestly all examples of the type. It's not particularly rich, but I'm glad I took the time to understand it. I knew none of it beforehand, and ended up deriving it all while trying and failing to construct puzzles where the parity would just not line up.

Every Second Turn has some similar structural deductions that are a bit harder to work out- but the type is also more generally approachable, so they don't come up as often. Consider this a bit of a teaser for the future...