How to Solve the Darts Puzzle
The darts puzzle is bad game design: In any game, a puzzle is expected to be solvable in some kind of way. Therefore hints exists. In the case of the darts puzzle, that hint is the dropped red arrow on the floor and the note in the drawer.
Those hints imply that the red dart was at some point attached to the board and fell down. This means you have to somehow figure out the original position.
As we cannot assume that there is a hint missing (it is on the ground after all) we can use our environment to try to derive that missing info from things immediately around us: The dart board. Here comes the design flaw:
The dart board offers an intuitive hint: A very prominent black pixel on the surface at the number 6:
There’s something off with the right side of the board – it’s the only part that looks wrong. This seems important, but when we try using the arrows, it doesn’t solve the puzzle. I remember there was a rules page back at the inn. Maybe we need to check that again, since we might be doing the math wrong. Looking at the 501 rules makes things more complex though, since there are different ways to count the score. Plus, we can put the 0 at the beginning or end of numbers, which means people think they’ve found the right clues but can’t figure out how they fit together.
Then you have a 4 digit code with a 3 digit solution giving a lot of ambiguity to any possible combination. Especially if you are following a wrong clue.
The maximum score with one dart without doubles, tripples and the bull (rings) is 20 (i.e. 20).
The maximum score with one dart without doubles, tripples and the bull is 25 (i.e. ring around bull).
The maximum score with one dart without tripples and the bull is 40 (i.e. D20 = 2×20).
The maximum score with one dart without tripples is 50 (i.e. bull).
The maximum score with one dart is 60 (i.e. T20 = 3×20).
36 can be obtained with one dart: D18 = 2×18 or T12 = 3×12.