(another riddle, not as long as the last)
The mysterious and shadowy contact who set you up with that formerly-haunted house was impressed enough with how you dealt with the infinite basement that they've got another job for you. There's this ballroom that's recently been de-ghosted, and you've been asked to join the team that's doing the initial post-deghosting inspection of it.
Hints
Maybe take another look at those tiles...
If you try to construct the pattern, what might you be able to do to make it work?
(aside from changing the shape, size, and/or arrangement of the tiles)
It's definitely impossible on a flat surface. Flat by real-world standards anyway.
Putting it on a sphere seems to do the opposite of what you're looking for.
If you don't recognise this here, it'll probably be very difficult to work out what's happening on your own! Don't worry, you can start looking stuff up.Solution
There's something quite unusual about the space in this room: a floor tiled with heptagons like that which lies "flat" can't exist in our world, but could exist in hyperbolic geometry. Compared to ordinary (Eucliean) geometry, hyperbolic space effectively has more space, letting you fit too many heptagons together. In a way, it's the counterpart to spherical geometry, where there's *less* space; hyperbolic surfaces have "negative curvature", while spherical ones have positive curvature.
Another property of the hyperbolic plane provides the answer as to how you're able to take two right turns without getting closer to your starting point. Honestly I do not actually know enough about hyperbolic geometry in formal terms to properly explain this, but uhh...
Polygons in hyperbolic geometry have interior angles that add up to less than you'd expect from Euclidean geometry. The larger the shape, the sharper the angles required. The path used in the puzzle is three sides of a square in Euclidean geometry (and would make a triangle on a sphere, if it was a small sphere), but on the hyperbolic plane you don't turn around far enough and end up making a path that looks more like a line. So, the second point is closer than the first!
Commentary (spoilers)
This was mostly inspired by HyperRogue, which is a game all about navigating a hyperbolic plane (and also how I made those images from before). HyperRogue has a feature where it shows your journey as an almost straight line at the end of a game no matter how much you remember turning about during it. If this was interesting at all, I recommend checking it out.
There's a similar riddle about geometry that asks how you can walk 10 miles south, 10 miles west, 10 miles north, and then end up in the same place. The answer is that you're at the north pole, though it's a bit sneaky because the "10 miles west" isn't actually in a straight line (you'd need to walk all the way to the equator for that). There's a version of this riddle that helps clue you in by using a bear and then asking what colour the bear is. The answer is that it's white because it's a polar bear, but apparently polar bears don't actually go near the north pole very often.
Then again, it's probably a bit unfair to expect all these riddles to be completely realistic.