u/No-Possible-263

Hello,

Can you solve all the puzzles given by the game and beat the minimax algorithm in this game derived by the Cayley tables of finite groups?

Here is the link: Tierisch Verbunden

The goal is to connect your animals orthogonally (diagonally does not count). Putting an animal a on the cell (b,c) with a = b*c , then your opponent in the next turn, can not move b and c. Thats the whole game 😄 Besides: 3 times repetition ends in draw.

Would be glad to get some feedback how difficult the puzzles are.

P.S: Each world with the background image of an animal, consists of some finite groups with same order. It starts with the trivial group and the C2 as a tutorial:

The trivial group: Who sets the first stone / animal, wins.

The C2 group: I will not spoil it, it is just a simple moment of reflection.

The C3 group: This puzzle is in the difficulty level of tic tac toe or a little bit above.

The C4 group and the Klein four group: Now starts the real puzzle.

It goes up to groups of order 10... and gets more interesting as the journey continues.

Lately I have been thinking about how to convert each finite group into a game ( puzzle, piece of art or music). Here I present a two player game on a Cayley table:

Put your animals orthogonally (diagonally not allowed) connected on the table or once they are on the Cayley table move them to get orthogonally connected. If you put h on h = f*g , then in the next move you block your oponnents pieces f and g. Situations which repeat three times result in remis. Players which have no mor legal moves lose.

Here is the game: Tierisch verbundene Welt (solo vs. Minimax - algorithm)

And here is a version to play it against a friend or yourself .

It is build like this: On the worldmap you see your groups (animals), you play against MiniMax algorithm level 2. Each solved world, opens you at least one next world (group, animals). It starts very easy with the trivial group. Who sets the first stone, wins. Then the second, with a moment of reflection it is also doable. The group C3 is a bit trickier and C4 or the Klein four group are difficult but doable. I have yet to solve C7 (managed to solve C5 and S3).

One can prove that the ratio of (number of solutions by white or black) / (number of legal figures in the game) goes exponetially to 0 as the group size goes to infinity, so expect each increase in group size to be more difficult to master.

It could be used as an exercise in Cayley tables or just for fun of solving a puzzle.

Any ideas on how to use group theory to construct a more efficient minimax algorithm with higher depth?

https://preview.redd.it/41t0ik7nvl1h1.png?width=1750&format=png&auto=webp&s=030257833bd2240fdfc34a62a1ff3c72015f641d

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Lately I have been thinking about how to convert each finite group into a game ( puzzle, piece of art or music). Here I present a two player game on a Cayley table:

Put your animals orthogonally (diagonally not allowed) connected on the table or once they are on the Cayley table move them to get orthogonally connected. If you put h on h = f*g , then in the next move you block your oponnents pieces f and g. Situations which repeat three times result in remis. Players which have no mor legal moves lose.

Here is the game. I guess it could be used as some sort of way to educate yourself and become familiar with Cayley tables, but what I fiind more interesting about to think is:

How do group theoretic properties reflect if one of the players have a winning strategy or not?

Does the game for group G allow advantage for the first / second player?

Here is the link to the game: Tierisch verbundene Welt

It is build like this: On the worldmap you see your groups (animals), you play against MiniMax algorithm level 2. Each solved world, opens you at least one next world (group, animals). It starts very easy with the trivial group. Who sets the first stone, wins. Then the second, with a moment of reflection it is also doable. The group C3 is a bit trickier and C4 or the Klein four group are difficult but doable. I have yet to solve C5.

One can prove that the ration of (number of solutions by white or black) / (number of legal figures in the game) goes exponetially to 0 as the group size goes to infinity, so expect each increase in group size to be more difficult to master.

Lately I have been thinking about how to convert each finite group into a game ( puzzle, piece of art or music). Here I present a two player game on a Cayley table:

Put your animals orthogonally (diagonally not allowed) connected on the table or once they are on the Cayley table move them to get orthogonally connected. If you put h on h = f*g , then in the next move you block your oponnents pieces f and g. Situations which repeat three times result in remis. Players which have no mor legal moves lose.

Here is the game. I guess it could be used as some sort of way to educate yourself and become familiar with Cayley tables, but what I fiind more interesting about to think is:

How do group theoretic properties reflect if one of the players have a winning strategy or not?

Does the game for group G allow advantage for the first / second player?

Here is the link to the game: Tierisch verbundene Welt

It is build like this: On the worldmap you see your groups (animals), you play against MiniMax algorithm level 2. Each solved world, opens you at least one next world (group, animals). It starts very easy with the trivial group. Who sets the first stone, wins. Then the second, with a moment of reflection it is also doable. The group C3 is a bit trickier and C4 or the Klein four group are difficult but doable. I have yet to solve C5.

One can prove that the ration of (number of solutions by white or black) / (number of legal figures in the game) goes exponetially to 0 as the group size goes to infinity, so expect each increase in group size to be more difficult to master.

The aim of the game is to construct a closed loop with as many tiles as you can.

For each small finite group G with n elements, I create n*n = n² tiles of n different types. The difficulty increases with n and the also more "complicated" groups tend to be more difficult to "solve". Default ist the Klein Four group V4, but you can change it easily in your browser below the board of the game. Any feedback, especially any ideas on how to make this to a fun multiplayer game? (Spoiler: For the group C5 we have maximal number of tiles: 25 = 5 * 5:)

https://preview.redd.it/9jowo20lgyzg1.png?width=1093&format=png&auto=webp&s=abfa85b6dd24e0637ff86203ecf54f7c82b248b9

Hello,

I wanted to share some music I have composed using finite groups :

MP3 rendered with DAW: https://www.orges-leka.de/music/abelian_group_2_9_5_three_voices.mp3

Score as PDF: I am not sure how readable the score is though, I have used musescore to expor the midi which is generated through an algorithm and sonifies the permutations of some finite group: https://www.orges-leka.de/music/abelian_group_2_9_5_three_voices.pdf

I have done more finite groups, but this one I like particularily and like to listen to while walking.