This is just a thought, but I wonder if there is a mathematical connection between this game and something like the binary representation of irrational (or maybe transcendental) numbers.
The article is also notable for its consistency in spelling "lose" as "loose".
One could interpret the outcome of the game as a number by ○ being the digit 0 and ● being 1. For fun we could also say that if there is a repeating subsequence at the end (someone lost), then that is repeated infinitely. I suggest this because any won game has a sub-string repeated three times at the end, so we might as well repeat it to infinity!
Say the example game, ● ● ○ ○ ● ● ○ ○ ● ● ○ ● ● ○ ● ○ ○ ● ● ○ ○ ● ● ○ ○ ●, would be 0.11001100110110100110011001, or perhaps 0.11001100110110(1001) where the parenthesis express infinite repetition. If we choose the first, it is 768160/959951 and the second would be 65553/81920.
In any case, a won game would be a rational number, while a game which goes on forever would be an irrational! One could then wonder which irrational numbers are represented by such games.
Just adding to op, for some (myself included) it is quite painful to see loose in place of lose, this should be fixed asap as it distracts the reader from the content. Lose = opposite of win, loose = opposite of tight.
Some irrational numbers are not valid games. For instance, I am sure the expansion of let us say π/4 in binary has 000 as a subsequence somewhere. But that could never happen in a game, because it disallows repetitions of a substring three times in a row.
Ah, you are right! At least wrt the "naive" mapping of irrationals to binary representation. My gut still tells me it's bijective, but the mapping has to be a bit more involved :D
The article is also notable for its consistency in spelling "lose" as "loose".