Strategy set in Tic-Tac-Toe I read in a book that the cardinality of the strategy set) of the first player in a game of Tic-Tac-Toe is approximately equal to $10^{126}$ but I cannot see how to arrive at this result. Disclaimer: I don' t want to calculate how many sub-games there are!

A strategy is a function defined from all possible positions to all possibles moves you can make from that position.

Note that there are $3^9$ positions in this game (at most) and 9 possibles moves at most there are at most $9^{3^9}$ possible such functions.

Of course much less because you can only consider the positions before you have to play that you can reach after previously applying the strategy so it's more like

$$9\times 7^{8}\times 5^{8\times 6}\times 3^{8\times 6\times 4}\times 1\approx 10^{132}$$

A little bit less because some games will end before filling the board.