If Tic Tac Toe can be computed/played by a a Tinker Toy computer I think it should be a doable thing for Turing Tumble. At its simplest the logic/pseudo code of Tic Tac Toe is:
If your opponent has two in a row - block it
else - play any square
Certainly this portion is possible in Turing Tumble. Reading the board and placing a piece in a square is certainly no trivial but with several Turing Tumbles and “Tumblers” working as a team it should work.
Here is a a mechanical Tic Tac Toe Machine that may lend some insights into our version:
The Tic-Tac-Toe Computer mk1: