how do you unlock the door that has a code in the start of the game?
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It's randomly generated on each playthrough. For clarity, it leads to the same location as the name guessing puzzle, so if you've already solved that one, you might not need to solve this one as well.
If you guess incorrectly, the game will give you more hints for how to do the math required to solve the puzzle.
The solution is (Number of turns) modulo (total number of notches) = (dial setting). Remember! The zero notch counts, so a 0-9 notched dial has 10 total notches. There are a few other ways to get to the right answer as well, but that's the easiest one to plug into a calculator.
(BTW, if you guess incorrectly a bunch of times your character will decide to just brute-force the combination, allowing you to pass the door without doing any math)