Prove that $5 \times 5 \times 5$ tic-tac-toe ends in a draw

I am pretty sure that when played perfectly, $5 \times 5 \times 5$ tic-tac-toe will end in a draw. Is anyone able to prove this?

Solution 1:

This seems to be an open problem. At any rate, it was stated as an open problem in József Beck's 2008 book Combinatorial Games: Tic-Tac-Toe Theory, on p. 55:

Open Problem 3.2 Is it true that $5^3$ Tic-Tac-Toe is a draw game? Is it true that $5^4$ Tic-Tac-Toe is a first player win?

Very little is known about the $n^d$ games when $d\ge3,$ especially about winning. We know that the first player can achieve a $4$-in-a-row first in the $3$-space ($4^3$ Tic-Tac-Toe); how about achieving a $5$-in-a-row? In other words, the first player wants a winning strategy in some $5^d$ Tic-Tac-Toe. Let $d_0$ denote the smallest dimension $d$ when the first player has a forced win in the $5^d$ game; how small is $d_0?$ (A famous result in Ramsey Theory, called the Hales–Jewett Theorem, see Section 7, guarantees that $d_0$ is finite.) The second question in Open Problem 3.2 suggests that $d_0=4,$ but what can we actually prove? Can we prove that $d_0\le1000?$ No, we cannot. Can we prove that $d_0\le1000^{1000}?$ No, we cannot. Can we prove that $d_0\le1000^{1000^{1000}}?$ No, we cannot prove that either. Even if we iterate this $1000$ times, we still cannot prove that this "$1000$-tower" is an upper bound on $d_0.$ Unfortunately, the best-known upper bound on $d_0$ is embarrassingly poor. For more about $d_0,$ see Section 7.

(See this old answer for a few more quotations from Professor Beck's book.)

If the answer to your question is known, it must have been found within the last few years. You might try asking Professor Beck himself.