These sequences are also known as "cubefree," so you might want to continue researching along those lines.
In particular, the game discussed is trying to find cubefree words over a two-letter alphabet. The sample infinite game seems to agree with the listed sequence on OEIS for the lexicographically earliest infinite cubefree word, though your method of generation appears to be different from the one in the comments. (I haven't analyzed it in detail.)
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.
There is a simple intuitive explanation for how an "infinite" game is possible:
We can define two different sequences of three characters that start with 0 and end with 1: 001 and 011. Because they each start and end with a different character, we can never create a series of three characters by chaining two of these sequences.
Now we can go one step deeper and encode the "001" sequence as 0, and the "011" sequence as 1. We can generate our 001 and 011 pattern again, but with our encoded versions of 0 and 1, giving us these sequences: 001001011 and 001011011. These sequences again have the same characteristics, they start and end with different sub-sequences (001 and 011) so they can be chained without creating a series of three sub-sequences.
We can now use these larger sequences and encode these as 0 and 1, and so forth ad infinitum.
I guess it also hints at why the sequences keep getting smaller when you compress it several times (as mentioned in a footnote). Each compression peels away layers in this expansion. That’s my intuition at least.
> The file is a mere 512 bytes, and unpacks to a 26kb file, which again unpacks to 3Mb.
My brain hurts when thinking about that. How could 512 bytes be enough to store ~3 million bytes? I know that compression is basically about finding patterns and this sequences should be very compressible.
If it was a file filled entirely with one character, the compression could simply be to write a file saying "this character copied 3 million times", which is less than 512 bytes.
This is not exactly what happens here, but many compression algorithms work by recognising that certain substrings are very common, and give them a "shorter code". In this game, there are some quite long such strings, giving a good compression rate. Furthermore, because of the recursive nature, it can find such patterns again after the common substrings are replaced by shorter codes, because these codes again form patterns with repeated substrings. This goes on until there is almost just a bit of meta data and an "ur-pattern".
Compression is fascinating in many ways. For instance, since there are a fixed number of files of a certain size and some bigger files are made smaller, some smaller files must be made bigger by the compression! Of course, this could be as simple as attaching a header or flag which says "I could not compress this. Here is the old file verbatim." But that is still a bit longer than the original!
This is the idea behind Kolmogorov complexity[0], that the complexity of a string (finite or infinite) can be measured, relative to a programming language, as the the length of shortest program which produces it.
Precisely computing the Kolmogorov complexity of a given string could be very difficult, though. In general, it is uncomputable because we cannot decide if a given program will output a given string.
You can think about the compressed size of some file as approximating the amount of information (in the Shannon sense[1]) there is in the file. A perfect compression would reduce the file to exactly the size of the amount of information it contains.
One relatively common way to remove first player advantage is to have the first player place one stone, and then after that each person places two stones on their turn. So
After player 1s first move there is one stone in the sequence.
After player 2s first move there are three stones.
After player 1s next move there are five stones.
After player 2s next move there are seven stones.
Etc. Usually this completely removes first player advantage. It’s obvious that this removes player 1s potential advantage since black or white, his move is symmetrical without loss of generality.
Player 2 actually has the first consequential move, with three possible options — again, taking player 1s move as a given color, player 2 can play
DD
DS
SD
where S means “same” and D means “different”. Technically player 2 has four options: he could also play SS and lose immediately :-)
My intuition would be that either there is an infinite game possible and perfect play produces a stalemate, or games must inevitably end, and someone must have a winning strategy — in which case I’m betting its player 2, who makes the first “real” move.
It's not intuitively clear to me how the game can go on forever -- I would expect that, eventually, you would hit upon some valid pattern. The explanation in the text didn't really make sense to me. Could anyone help with this?
The game could use a better definition of what constitutes a pattern.
>> A pattern is a sequence of pebbles repeated three times in a row.
By that logic, player 2 would have lost at their fifth turn. ... And 'in a row' is open to interpretation since it is additional information to 'repeated three times'.
I think it's pretty clear that a string s of pebbles is a pattern iff there exist strings x and y such that s = x|y|y|y, where | denotes string concatenation and y is nonempty.
This reminds me of the movie "The Oxford Murders" with Elijah Wood, where a maths professor and their student argue if any pattern can be predicted by logic. Well worth a watch.
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In particular, the game discussed is trying to find cubefree words over a two-letter alphabet. The sample infinite game seems to agree with the listed sequence on OEIS for the lexicographically earliest infinite cubefree word, though your method of generation appears to be different from the one in the comments. (I haven't analyzed it in detail.)
https://oeis.org/A282317
From your link it seems it is a conjecture that this is the lexicographically earliest one. Very interesting!
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.
Without having a proof ready at hand, I am quite sure that the `generate` sequence from my post represents a transcendental number.
Seems like there should be a bijection there no?
We can define two different sequences of three characters that start with 0 and end with 1: 001 and 011. Because they each start and end with a different character, we can never create a series of three characters by chaining two of these sequences.
Now we can go one step deeper and encode the "001" sequence as 0, and the "011" sequence as 1. We can generate our 001 and 011 pattern again, but with our encoded versions of 0 and 1, giving us these sequences: 001001011 and 001011011. These sequences again have the same characteristics, they start and end with different sub-sequences (001 and 011) so they can be chained without creating a series of three sub-sequences.
We can now use these larger sequences and encode these as 0 and 1, and so forth ad infinitum.
I guess it also hints at why the sequences keep getting smaller when you compress it several times (as mentioned in a footnote). Each compression peels away layers in this expansion. That’s my intuition at least.
My brain hurts when thinking about that. How could 512 bytes be enough to store ~3 million bytes? I know that compression is basically about finding patterns and this sequences should be very compressible.
This is not exactly what happens here, but many compression algorithms work by recognising that certain substrings are very common, and give them a "shorter code". In this game, there are some quite long such strings, giving a good compression rate. Furthermore, because of the recursive nature, it can find such patterns again after the common substrings are replaced by shorter codes, because these codes again form patterns with repeated substrings. This goes on until there is almost just a bit of meta data and an "ur-pattern".
Compression is fascinating in many ways. For instance, since there are a fixed number of files of a certain size and some bigger files are made smaller, some smaller files must be made bigger by the compression! Of course, this could be as simple as attaching a header or flag which says "I could not compress this. Here is the old file verbatim." But that is still a bit longer than the original!
Precisely computing the Kolmogorov complexity of a given string could be very difficult, though. In general, it is uncomputable because we cannot decide if a given program will output a given string.
[0]: https://en.wikipedia.org/wiki/Kolmogorov_complexity
[1] https://arxiv.org/pdf/1612.09316
Deleted Comment
After player 1s first move there is one stone in the sequence.
After player 2s first move there are three stones.
After player 1s next move there are five stones.
After player 2s next move there are seven stones.
Etc. Usually this completely removes first player advantage. It’s obvious that this removes player 1s potential advantage since black or white, his move is symmetrical without loss of generality.
Player 2 actually has the first consequential move, with three possible options — again, taking player 1s move as a given color, player 2 can play
DD DS SD
where S means “same” and D means “different”. Technically player 2 has four options: he could also play SS and lose immediately :-)
Would your intuition be that this makes it so that neither player has a winning strategy? That perfect play would yield an infinite sequence?
>> A pattern is a sequence of pebbles repeated three times in a row.
By that logic, player 2 would have lost at their fifth turn. ... And 'in a row' is open to interpretation since it is additional information to 'repeated three times'.
EDIT: Clarify that only y is nonempty.
Could you tell us more about this? I am curious how this problem was formulated using modal logic. Seems fascinating