.. How do I open the door that looks like a dartboard?

Magnus's note says that the numbers show how many sections that touch that piece have been "filled in." Read to the end of this hint for one way to do this.

To fill in a piece, click on it. The numbered pieces can't be clicked on.

The sections that touch include the ones that only touch on the corners.

You can only click on 10 pieces, as shown by the lights at the left. This means you'll want to have each piece touch as many numbered pieces as you can.

There's more than one way to do this, especially in the amateur puzzle, but here's what worked for me:

Secret Room Puzzle 1, Amateur

Secret Room Puzzle 1, Master

The puzzle slides up to reveal another puzzle, this one with gears.

Where do I find all the gears I need?