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Oh my god, I fell RIGHT INTO IT. That's actually genius xD. Honestly, I think you did it the right way, that's on me xD. I just fell into the trap of thinking that this door was just the kind of bullshit level you throw 1h before the deadline of a game jam, but it was way more genius than that xD. Maybe, adding even more buttons, so there is a ridiculous amount everywhere on the wall, could make the player understand better, but honestly
I finished it and got the cake, but it was a lie ='( xD. The "written numbers on the wall" puzzle was a bit tricky, it took me many tries to find the correct order of the numbers... I don't know if there was a logic to it or not xD.
I adjusted my ratings accordingly, because I feel so stupid to not even try to open the door that you deserve one more "Game design" star xD.
Awesome game, keep up the good work ;).