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Tic-tac-toe can be certainly won in 2D (on a 2-dimensional square grid including diagonals) already if the field is just slightly larger: 4x4 (while 3x4 still is a draw).

There is an excellent video by 'PBS infinite series' -see references- that looks into longer lines needed in a grid of larger dimensions for the game to be a certain win or draw.

I wonder about tic-tac-toe on a graph: What about $t^3$, 3 adjacent tics or crosses needed, on a simple graph (no loops, no multi-edges)?

What is the condition on the graph for the $t^3$ game (3 adjacent tics needed) to be a certain draw or a certain win?

  • On a complete graph with 5 or more vertices, you can always win...
  • On a cycle graph, you can always draw

References

Michael T
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    You probably should use other word instead of "connected". In graph theory, two vertices are connected when there is a path from one to the other. So in any connected graph with 5 or more vertices you can always win. You're probably thinking in the vertices being adjacent. – jjagmath Mar 09 '25 at 11:40
  • @jj, thx for the correction – Michael T Mar 09 '25 at 13:11
  • See https://math.stackexchange.com/questions/5050304/on-which-simple-graphs-no-loops-no-multi-edges-is-the-tic-tac-toe-variation-t for a variation of this where the player wins that tics a triangle... – Michael T Apr 27 '25 at 15:11

2 Answers2

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The problem is solved in Robert A Beeler, Tic-tac-toe on graphs, Australasian Journal of Combinatorics 72(1) (2018) 106-112, available at https://ajc.maths.uq.edu.au/pdf/72/ajc_v72_p106.pdf

The 1st player has a winning strategy if and only if the graph contains a subgraph of the form $\{12,13,14,15\}$ or $\{12,13,14,35,45\}$ or $\{12,13,14,35,46\}$ or [$\{12,13,45,46\}$ and a path with an odd number of vertices running from $1$ to $4$]. On all other graphs, both players have a drawing strategy.

Gerry Myerson
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  • many thanks, that article was on my reading list - now it definitely moved up! Any references to a variation of the game to tic-tac-triangle or sic-sac-soc-square? – Michael T Mar 09 '25 at 12:53
  • Sorry, I don't know, I didn't look for any. – Gerry Myerson Mar 09 '25 at 21:37
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    jfyi: a presentation version of Beeler's article is available here https://faculty.etsu.edu/beelerr/t3-talk.pdf – Michael T Mar 24 '25 at 06:47
  • Why doesn't a standard 3×3 board have a subgraph of the first type, with "1" as the central square? – MJD Mar 27 '25 at 07:37
  • I see now: in this model, the winning configurations on the 3×3 board include (top left, top center, center). – MJD Mar 27 '25 at 07:41
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Not an answer, just a screenshot to a presentation from Beeler showing the subgraphs that allow winning strategies:

enter image description here

References

Michael T
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