Higher Dimenional Tic Tac Toe Here we have a problem that seems very intuitive, but is hard to define mathematically.
In Tic Tac Toe, we can find an equivalent of the game in any number of dimensions, it seems.
The trick is to realize the the game is defined by lines on which the characters can be placed not how the board is drawn.
1D tic tac toe, for example, would be a single column of boxes.
2D tic tac toe, is the classic childrens game.
3D tic tac toe looks like this: 
(Thanks to PrintActivities.com for the image, which I've shrunk.)
Now, here is the pattern I've noticed:
1-dimensional:


*

*up/down


2-dimensional


*

*up/down

*left/right

*diagonal


3-dimensional


*

*up/down

*left/right

*diagonal

*vertical up down

*vertical diagonal


The number of possible winning combinations seem to be $2n-1$!
Now... how would I express this for the fourth spatial dimension?  What ways can this be projected so some one with a worse short-term memory than Leonhard Euler can actually play it ?
This sounds fun (if challanging) to analyze myself, but the question is complicated enough I am interested in the insights of others!
Also, this question is to help assemble a computer game (I'm a programmer) but I intend to link back at least to this page from the game itself, credit where credit's due.

The list of winning combinations does not distinguish between e.g. top-left to bottom right diagonal and top-right to bottom left. The $2n-1$ comes from $n$ axes and the $n-1$ kinds of digonals that occur when each box in the line is shifted from the previous along $k$ axes, where $2\leq k \leq n$.
The $4$D board could be represented by three $3$D boards, each is which is represented by three $2$D boards.
