I would invest in a cheap scratch pad. The kind with pages held together by that rubbery glue stuff. And a 10-pack of Bic or otherwise "cheap, but decent" pens. And write stuff down.
One thing I can tell you is the chess pieces (spoiler tier 1)
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don't really matter in each run
(spoiler tier 2)
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They matter in one run
(spoiler tier 3)
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They matter if you unlock a certain room that has a chess puzzle in it
(spoiler tier solution)
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You will need to know where certain chess pieces are that day in order to solve the puzzle. The pieces are assigned to a room, not a grid location. A room with a pawn will always have a pawn, and you should know where on the 9x5 grid that pawn is.
That said, I haven't gotten far enough to do the chess puzzle yet. I've just, um, heard things.