Explore the hypercube as a tictactoe setting. Tictactoe is a simple game with the goal of getting three squares in a row (or diagonally).
On a one-dimensional board of three squares, it would be impossible to win if players took turns.
The conventional, two-dimensional board of nine squares provides one of the earliest games a child might learn to play. But once players become familiar with the layout, a cat's game (where no one wins) becomes standard.
I've bypassed the three-dimensional, cubic gameboard to go on to the four-dimensional hypercube. With rows and diagonals going off in many dimensions, the game becomes interesting again, at least for a while. There's too much of an advantage to going first, though, because one can set up threats in two or three dimensions at once.
In order to reduce the first player's advantage, I've introduced blocking with neutral squares at higher difficulty levels.
Later versions may allow the user to turn the hypercubic board in order to more clearly see rows and diagonals that look widely separated on the static board. Perhaps there will also be a playe vs. player mode.