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So I was wondering in tic tac toe in infinite dimensions where instead of a 2D grid I play on an infinite dimensional hypercube. I've come to the conclusion the term 'optimal strategy' does not make sense (?).
How? - Imagine I score a win using an optimal strategy (assuming it exists). Then using the projection operator I can project part of the game on a 2 dimensional grid and work backwards, then add a cross product of another 2d grid and so on...

What I mean by 'optimal strategy' is one cannot even make statements like corners are the optimal positions. Consider the counterexample in the simplest 2d case (normal tic-tac-toe with chess notation).

X: B1 O: A3 X: C3

Now O is forced to move A1 (anything else loses)...
So, now one may add forced moves + corners are a good thing strategically...

My question is:

How does one prove or disprove an optimal strategy exists/does not exist? How does one define complexity of the optimal strategy?
(Assuming that's a sensible question)

Michael T
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More Anonymous
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  • If an optimal strategy does exist then what makes this game special that one can clearly define it? – More Anonymous Mar 16 '24 at 06:32
  • How many in a row do you need to win? – fleablood Mar 16 '24 at 06:37
  • I'm tempted to say more the merrier but even in the case of one I'd like to see this play out ... – More Anonymous Mar 16 '24 at 06:40
  • @fleablood I'm sorry I thought you said how many rows do you need to win. 3 in a row should be enough. – More Anonymous Mar 16 '24 at 06:45
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    Have you looked into finite-dimensional tic-tac-toe strategies?According to this answer the first player has a winning strategy on a $3^n$ board if $n\ge 3$. You could try applying the same strategy to the $3^\omega$ game. – Karl Mar 16 '24 at 06:51
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    Well, a winning strategy for 3 exists for 3 in a row if the dimensions are two or more and the size of a board is infinite (or, I think 4 or more). The first player can always win. Move 1: make a mark. Player B does any thing. Move 2. Make a mark adjecent to the original mark in such a way that you have two in a row unblocked on either end. (If you had $n$ dimensions there are $3^n-1$ adjecent neighbors. Player Bs mark can cut of $2$ of them but there are still $3^n-3$ choices). Player B can only block at most one. Move 3: Put a mark in an open end and get three in a row. – fleablood Mar 16 '24 at 16:01

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