Question 70
How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$? (That is, how many lines pass through three of the lattice points $(a_{1}, \ldots, a_{n})$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\}$?) Express your answer in terms of $n$.