Everything we know about the Billiard Room Dartboard Puzzle in Blue Prince

Mt. Holly has tons of mysteries in Blue Prince, but few of them beat the early-game challenges of the dartboard puzzle in the Billiard Room. It's not really required if you want to reach the elusive Room 46, but this math mini-game yields great rewards such as randomized keys to open locked pathways, shortcutting your progress in future runs.
Here's everything we know on solving the dartboard puzzle with some good reasons to invest time in it.
Billiards? Hardly any of those take place in the Billiard Room. More of a mark for the dartboard, and even more on the colorful mathematical symbols that decorate its periphery. Each dartboard puzzle consists of a stack of equations based on dartboard rings and color-coded operations.
Color code sheet:

Here's the kicker: normal math rules don't apply here. In Blue Prince, we work from the center outwards beginning with the ring nearest to the bull's-eye.
And here's a simpler puzzle just to illustrate.

Around your tenth or fifteenth visit there will be odd, really weird symbols in the bull's-eye or anywhere in the segments of the dart. Those are just modifiers after your main equation, and mastering those will enable you to move through the puzzles in the late game.
These can also stack. So yellow Square and Diamond would imply subtract, square, and reverse.

Pro Tip: Your final answer must be between 1 and 20 because all intermediate results can actually be negative or decimal. Just trust it—work outward carefully, and always double-check symbol effects before locking in an answer.
A correct solution earns random keys. That's at least 1 key 'cause they will come from the most populous standard door key to its very rare finds like:
It helps open both locked doors and possible alternate paths to rooms that might contain lore, loot, or cutting corners.
The billiard room dart puzzle in blue prince has been such an incredibly fine mechanism: logic, memory, and experimentation all put together. It is a tiny slice of the whole design of the game: cryptic yet fair, mysterious but rewarding.
So the next time you trudge through Mt. Holly, don't miss out on the dartboard. Get up there and crunch the numbers to win that prize.