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That's a neat thought! 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
by gylterud 1y ago
That's a neat thought!
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.
- jpkw 1y agoJust 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.
- deleted 1y ago[deleted]
- gylterud 1y agoShould be fixed now! I hope the writing is easier to bare now.
- ajd555 1y agoA few other typos (first sentence, though instead of thought, trice instead of thrice, simlpe instead of simple). Hope this helps!
- soulofmischief 1y agoAlso, weather -> whether
- deleted 1y ago[deleted]
- fxwin 1y ago> One could then wonder which irrational numbers are represented by such games. Seems like there should be a bijection there no?
- gylterud 1y agoSome 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.
- fxwin 1y agoAh, 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
- deleted 1y ago[deleted]
- gylterud 1y agoA neat, but perhaps difficult question just occurred to me: Is every irrational number which is represented by an anti-pattern game transcendental? Without having a proof ready at hand, I am quite sure that the `generate` sequence from my post represents a transcendental number.