Four-dimensional tic-tac-toe
Tic-tac-toe is a classical game; however, in the standard two-dimensional version of the game, it is far too easy to guarantee a draw or a win. To make the game more interesting, two players decide to play on a four-dimensional grid. How many different ways are there to make a line of 3?
The answer is 272.
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1 solution
The simplest way to count the lines on an n -dimensional grid (which I stole from this answer on StackExchange ) is to build an extra layer around it, so that it's the center of a 5 × 5 × ⋯ × 5 hypercube; and now the key observation is that every winning tic-tac-toe line intersects a pair of points on the outer layer, and every point on the outer layer intersects exactly one winning tic-tac-toe line on the inner hypercube. (The easiest way to see this for me is to put coordinates on everything and then see that each coordinate of the outer point uniquely determines the corresponding coordinate of each point on the line.)
The number of points on the outer layer is 5 n − 3 n , so the number of lines is 2 5 n − 3 n . When n = 4 this gives 2 7 2 .