Interesting! I think there is a lot of math going on in these kind of games. I have noticed interesting patterns in my game https://videopuzzle.org/ as well.

In daily mode, everyone gets a video scrambled into 4x4 pieces, in the same way, and needs to unscramble it. The operations available to unscramble a video are swapping a tile, and shifting the entire board. The lowest score every day is usually 12-14, but sometimes much lower.

Before I introduced the shifting function, surprisingly, every puzzle could only be solved in an either even or odd number of moves. Some days, all solutions would be 12-14-16 etc, while other days only 13-15-17 were possible. The arrows have changed this, as you can sometimes shave another move off if you use it smartly.