Tic-Tac-Toe to the Nth Dimension

In a normal \(3 \times 3\) tic-tac-toe board, there are 8 winning lines.

How many winning lines are there in a \(4\times 4\times 4\) tic-tac-toe board? (In this case, a winning line consists of four boxes in a row, either on the surface or inside the cube.)


Bonus: How many winning lines are there in a \(n^d\) tic-tac-toe hypercube, where \(n\) is the number of cells per side and \(d\) is the number of dimensions?

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