Really fun puzzle game that I had a blast playing (not just the puzzles, but the art was very visually appealing). The few issues I have with the game have all already been addressed below. Only things I would like is a level select screen or a restart button to re-try certain levels and some sort of fail condition. Theoretically, you can could complete every level by cutting everything into a 1x1 square and filling up the pattern. Rather than just suggesting players to do fewer cuts, maybe implement a maximum cut number (doesn't have to be the minimum required to complete the level) to bring further challenge to the puzzle.
Overall a game I thoroughly enjoyed as a puzzle game enthusiast, and I can definitely see what you mean by "our games went down very similar paths" haha.