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My other theory was that the colours were relevant but then only half of it made sense to me so I disregarded them. Can you tell me how to solve it?

The game is bugged. The Square goes in the Square.

Those are all the clues you need.

(+1)

I've since updated the game, but can't upload the new version while voting is in progress. While I don't want to outright leave the solution to the puzzle in the comments below my game, I will at least leave you with the additional hints I added. In the new version, there is now dialogue where Amelia tells Noah that they already tried 1) matching the shapes to the holes and 2) ignoring the puzzle entirely. From this, Noah gathers that he must *not* match the shapes to the holes, and cannot ignore the puzzle.