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Answer:
It has nothing to do with the pool balls, pool table, or darts. It's all about math (but not correct mathematical order of operations) and knowing what the colors mean on the dart board.
Blue indicates just that number (i.e. blue on 3 means press 3)
Yellow means you need to subtract that number from the Blue number that is indicated (i.e. Blue on 15, Yellow on 4 = 11)
Red means multiply.
Purple means you divide.
Do not follow the mathematical order of operation (PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).
With this in mind you should now be able to puzzle out the correct solution to the dart game. You've got this!
I will make a new run of the game and look again because the only color I remember seeing was the blue on the dart board and the sides being empty but I probably missed it.
Based on the info you're mentioning, I think the note is causing a lot of the confusion and might need to be re-worded a little bit. I'll try again, thank you both.
The note mentions a new game and the billiards room twice, yes it's IN the billiards room but calling attention to room led me to believe the billiards table was part of the puzzle.
I admit, part of that was me over thinking the puzzle but was playing while talking to a friend and they also thought, based on the note, that the billiards table was part of the puzzle.