Syzygy

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As of Feb 24 2022, Syzygy was dropped from its probably-too-high pricetag of $20 USD to "Free to Play" -- I can't see why I wouldn't recommend at least trying it if the basic idea you see on the store page appeals to you. You've got nothing to lose, and thar be at least ~5+ hours' of unique gameplay in them thar hills.
You can click on your frog self to cycle between squares, hexes, and triangles as the basic shape of cells that form the board on each level, and then you can click on edge-sharing adjacent cells to move to them, making your way to the goal. The idea is that if you cycle your cell-shapes right, you'll be able to open and close paths to weasel past enemies, or jiggle the positioning of objects you can kick around the board that you can use to lock up enemies or to hold down buttons, etc. I myself have enjoyed the first ~25% of the game, but I now feel like I'm not quite catching some of the logic, and that's pushing me into brute-force "let me try this and get nowhere" loops. Then again, the overworld map is fairly disorganized, so perhaps I've wandered into harder levels and just don't know it?
If you do try it I would recommend that you keep your expectations low in terms of functionality, quality of life, and presentation. It has undo and reset, but beyond that things are limited. I was unable to launch the game properly using the default settings on the odd pop-up launcher, and had to fiddle with them. My first play session I could only get going in windowed mode; I got fullscreen to work later. The game offers zero instructions on how to play -- in the trailer you can see the first level, where a sign is posted that invites you click your mouse: that's it. Think you're clever, do you, and you'll mosey over to the settings menu to get a leg up on the control scheme? Nay; many of the keyboard controls that are listed in the settings menu don't appear to work properly or else I don't understand them -- not sure how that could be the case. So, click I did, and I just had to feel like a dope for a bit until I worked it out. You'll do the same. The game has only 2 languages, Italian and English, and the English transliteration is at times either incoherent or just so opaque in its intended meaning that cannot tell which. The only audio setting available is a single slider for volume. The background music would be ok except that the decision was made to warp (speed up or slow down) the track depending on which shape you're currently displaying, so that the music becomes very irritating and disorienting, constantly changing as you cycle through shapes. It would have been nice to provide the option to toggle that off and just let music play normally. I've only heard limited sound effects so far, yet they've already proven problematic. For example, some levels have "icy" surfaces, so upon moving you (and NPC guards) zip around until you hit an obstacle; the same quick sound effect repetitively triggers every time you slide over a square/hex/triangle; this becomes rather grating.
That's all feedback on functionality and quality of life that I would be offering to the Dev (along with also supporting things others have suggested) if they were still interested in improving its usability and presentation. (Maybe they are interested; I don't know.) I'd also be suggesting specific tweaks to the control scheme (e.g., get keyboard controls functioning properly to supplement mouse; on "icy" levels, let me click on distant cells I want to slide to and just move me there, rather than requiring me to click on the cell adjacent to me). But since it doesn't seem like Syzygy is still undergoing revision, I can only offer these comments to you, dear reader. Even if the design were still in flux, it'd probably be too late to suggest doing much inside the game itself either in terms of narrative or in terms of art to make the world somewhat coherent: the art is fine, but... why am I a frog? why are pig-soldiers after me? why am I kicking around turtle shells? where am I? Where am I going?
I can't help but offer a quick "post mortem" set of guesses as to why Syzygy didn't find its audience. First, I think the Title of the game didn't really help it jump out at anyone. I'm pretty well-read, but I don't think I had heard the word before, and even now that I know it, the hand-drawn "title card" shown on Steam with the S, Z, and G integrated into Ys is, to me, still pretty illegible. It's hard to catch eyes when people don't know what they're looking at. Second, supposing you got past that and clicked to the store page, the minimal trailer and the bare-bones description -- which does little more than to call the game "enigmatic" -- didn't communicate much and probabaly didn't help to hold attention. That could've been a place to lay out some narrative and provide some sense of coherence and direction to the world/game. Third, of course, was the original pricetag of $20 USD -- which probably felt completely unjustified to many window-shoppers in light of the Second point just above. The store page feels like a big black box with a high price tag on it: what's in it? Fourth, there were very few pre-Free-to-Play reviews to provide information where the store page didn't, and not all of those were terribly helpful.
Maybe a demo would have helped. I wish that I had gotten to my review copy earlier. I'll reiterate the First and Second points above -- I had very little idea of what the game really was -- but I'll also pretend my review might have made a difference and take some blame here: maybe I could've done more to help bring attention to the game, and I definitely could have provided feedback that, I think, would have helped make it more pleasant to play. I would never have recommended the game at $20 USD, but for a more modest price tag and with some good faith efforts to improve the weaker points of design and presentation, I think Syzygy could have fared much better -- I still think it could, if the Dev has any interest.

I'm really impressed by this game!
I've been aware of this game for some time now, and was worried the gameplay might be too limited, that it would end up like a lot of indie puzzle games, with bigger and more trial and error type complicated levels instead of more challenging puzzles, but I was totally wrong!
I'm amazed at how this game keeps feeling fresh as I progress (I'm currently at 76% completed), always bringing in new game systems and new ideas. And there hasn't been a single level so far that I've solved using trial and error. All the puzzles are solved by analysing the level and the way the three tile shapes and other game systems affect the gameplay. Another great thing is that the challenge always comes from the great puzzle design and not from overly big and complicated levels as is too often the case in indie puzzle games.
The art-style and music might not be for everyone, but I personally love it! It fits very well with the gameplay, and it feels really polished, with the music and even the UI changing as you change the tiles shape.
Another great thing, that is rare these days, you have access to multiple save files from the menu!
The only thing I would like to see improved is the controls when playing with a controller. Currently, to move you have to select a tile using the d-pad or joystick, and then click on A. You get used to it really quickly, so it's definitely not a deal-breaker, but it would be nice if we could move directly using the joystick without needing to select and then click.
Overall, I'm really enjoying the game! Amazing puzzle design, game design and art. A really refreshing game!

nice puzzle, i was stuck.
Developer could add a dlc, something like developer support pack.
So for those who tried this game for free still may support with some money

Interesting puzzle game with unique mechanics, definitely worth checking out. It's relatively short, but polished and intriguing, reminds me somewhat of yankai's pyramid or hyperrogue (mostly due to the odd ways you interact with shifting geometry).

First of all I generally like puzzle games that are a bit different.
In Syzygy you shift the floor between the shape of triangles, squares and hexagons. It is a movement game and with the change of the floor other fields are connected/seperated. Obviously triangular fields are connected to 3 other and hexagons to 6 other fields. By changing you can evade enemies or move objects in a different way.
I am not the biggest fan of the music in this game and it could be looking a bit better.
All in all you get ~ 6 hour of puzzles with the game.

An amazing effort for a first game by Riccardo Caprari. An excellent puzzle game, where the concept of 2D tessellations is used to its full extent.
Outside of a few hiccups towards the end, where I found little to no intuition in the puzzles (and had to retract to a Monte Carlo approach to solve them by doing random moves, possibly because I was getting tired though), all the other ideas throughout the rest of the game were excellent.
It is regrettable that the game didn't do well, as it seems to have everything a game needs to do well. Hope Riccardo continues to develop games, he really did a great job with Syzygy, a far better job than most first projects out there.

Free puzzle game with some very hard puzzles that feels like a mix of many other puzzles in order to make something really unique. It consists of moving around in a grid that you can at anytime switch between an hexagon grid, to a square grid to a triangle grid, each type of tile you're on completely changes the way you move around.
Even as you're moving around the world to find the next stage (in an open-world environment much like in Stephen's Sausage Roll), you're trying to figure out what is best path and tile type you wanna walk on.
It has so many elements and mechanics that are slowly added to the game in a way that is smooth and not overwhelming (although, I'll say it again: This game is HARD!). It reminds of Sokoban, Chinese checkers, chess, that fox and chickens game, and probably some others.
Guaranteed free fun por puzzle lovers!