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Whereas pencil and paper play almost always involves just two players, Parker Brothers' rules said that up to three players could play.
The circular playing pieces resembled small poker chips in red, blue, and yellow. The 3x3x3 version of the game cannot end in a draw and is easily won by the first player unless a rule is adopted that prevents the first player from taking the center cell.
In that case, the game is easily won by the second player. By banning the use of the center cell altogether, the game is easily won by the first player.
By including a 3rd player, the perfect game will be played out to a draw. By including stochasticity in the choosing of the side the player must use, the game becomes fair and winnable by all players but is subject to chance.
By making the choice of the player piece x or o subject to chance, the game becomes fair and winnable by all players. On the 4x4x4 board, there are 76 winning lines.
On each of the four 4x4 boards, or horizontal planes, there are four columns, four rows, and two diagonals, accounting for 40 lines.
There are 16 vertical lines, each ascending from a cell on the bottom board through the corresponding cells on the other boards.
There are eight vertically-oriented planes parallel to the sides of the boards, each of these adding two more diagonals the horizontal and vertical lines of these planes have already been counted.
Finally, there are two vertically-oriented planes that include the diagonal lines of the 4x4 boards, and each of these contributes two more diagonal lines—each of these including two corners and two internal cells.
The 16 cells lying on these latter four lines that is, the eight corner cells and eight internal cells are each included in seven different winning lines; the other 48 cells 24 face cells and 24 edge cells are each included in four winning lines.
The corner cells and the internal cells are actually equivalent via an automorphism ; likewise for face and edge cells.
The group of automorphisms of the game contains automorphisms. It is made up of combinations of the usual rotations and reflections that reorient or reflect the cube, plus two that scramble the order of cells on each line.
Good work! Just a question, Are you going to go into video games? I assume you'd do well there. Your work is just phenomenal. I used to do games almost exclusively, but I discovered that programs that are largely technical rather than graphical are more fun.
Simulations are cool too. You can call me your first and biggest fan. Your games have reached cereal box quality. People also like.
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