Higher-Dimensional Tic-Tac-Toe | Infinite Series
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when discussing winning strategies, draws and line counts for tic-tac-toe on wider and higher-dimensional boards.
Recommended by 1 people across 1 different months, 2017 to 2025.
What it is
A PBS Infinite Series episode hosted by Kelsey Houston-Edwards on generalizing tic-tac-toe by widening the board, adding dimensions, or both. It covers when a first-player win or a forced draw is possible, drawing on the Hales-Jewett theorem and related papers.
Cited in 3 Wikipedia articles, including List of unsolved problems in mathematics, Hales–Jewett theorem, Nd game.
What people said
Recommends orlp · 2017-12 · Mathematics Stack Exchange · on “Is there a dimension of tic-tac-toe where there can never be a draw game?”
For the 3 by 3 board in 3 dimensions, the first player will always win with optimal play. Generalizations of $n$ by $n$ boards in $d$ dimensions is explored in this very nice video by PBS Infinite Series [this video] .
Cites Rishabh Jain · 2019-09 · Mathematics Stack Exchange · on “Number of lines in a tic tac toe of width $w$ and dimension $d$”
I was watching a video by pbs infinite series on tic-tac-toe( [this video] ). Here, they discuss if there is a winning strategy for the starting player in tic-tac-toe of dimension $d$ and width $w$ . I was trying to calculate the number of lines that are possible in a board, i.e, no. of lines that can result in a win for a board of arbitrary dimension and width. I want a formula which gives No. of lines $= f(d,w)$ . For example, a $3*3$ tic tac toe has $8$ winning lines. For $2$ dimension, it is easy to see that this number is $2*(w+1) , w$ horizontal, $w$ vertical and $2$ diagonal. But I couldn't generalize this to arbitrary dimensions.
Cites Michael T · 2025-03 · Mathematics Stack Exchange · on “On which simple graphs (no loops, no multi-edges) is tic-tac-toe winnable, where is it a draw?”
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
Cites Michael T · 2025-03 · Mathematics Stack Exchange · on “On which simple graphs (no loops, no multi-edges) is the tic-tac-toe variation t-t-triangle (aka cyc-cac-cycle) winnable?”
Tic-tac-toe ( $t^3$ ) can be played on a graph. It is very similar to tic-tac-toe 2 players; one X, one O X starts, then O, repeat the player, who can mark 3 adjacent vertices ('tics') in the shape of a 3-path ( $P_3$ ), wins The structure of the graphs allowing a winning strategy is known. Basically, all subgraphs allowing a winning strategy have been found - see On which simple graphs (no loops, no multi-edges) is tic-tac-toe winnable, where is it a draw? [LINK] . The $t^2$ game (2 adjacent vertices needed) is very easy; it is winnable, if the graph includes a $P_3$ subgraph (path with 3 vertices and 2 edges). This is a bit boring as then only the singleton and $P_2$ graph lead to a draw..