Qubic

Qubic is the brand name of a four-in-a-row game played in a 4x4x4 matrix sold by Parker Brothers starting in 1953. The original box, and the 1972 reissue, described the game as "Parker Brothers 3D Tic Tac Toe Game". Players take turn placing pieces to get four in a row horizontally or diagonally on a single board - or vertically in a column or diagonal line across four boards.

The game can also be played with pencil and paper. In the 1970s 3M Games (a division of 3M Corporation) sold a series of "Paper Games", including "3 Dimensional Tic Tac Toe". Buyers received a pad of 50 sheets with preprinted game boards.

Construction

The four boards were made of clear plastic (in a simple square design in the original release and in a funkier design for the 1972 reissue) with circular playing pieces that resembled small poker chips in red, blue, and yellow; each player used a single color. Markers could be placed in any unoccupied position, rather than stacked in a pile on a square as in Score Four. The game is no longer manufactured.

Game play and analysis

Either two or three players could participate in a game. The players take turns adding their own markers to the board. The first player to achieve four of their own markers in a row wins.

There are 76 winning lines. The 16 positions lying at the 4 space diagonals (8 corners and 8 internal positions) are equivalent and each involved in 7 winning lines; the other 48 positions (24 face positions and 24 edge positions) are also equivalent, each being involved in four winning lines. (The equivalence of a corner and an internal position is via an inversion; likewise for a face and an edge position.)

All of the analyses described here are for the two-player version of the game:

Qubic was weakly solved by Eugene Mahalko in 1976, establishing that in two-person play, the first player will win if there are two optimal players. A more complete analysis, including a complete first-player-win strategy, was published by Oren Patashnik in 1980. Patashnik used a computer-assisted proof that consumed 1500 hours of computer time.

The game was solved again by Victor Allis using proof-number search.

A more general analysis of Tic-Tac-Toe-like games, including Qubic, appears in Combinatorial Games: Tic-Tac-Toe Theory by József Beck.

Fool's Mate strategy

The cube structure makes the 8 corner-points and 8 centre-points extremely important; each of these is a member of 6 planes (1xflat, 2xvertical, 2xdiagonal-vertical, 1xcross-vertical) of 16 points. The flat and 2xvertical planes only contain 4 important points, while the 2xdiagonal-vertical and cross-vertical contain 8 powerpoints.

O begins and places his first peg A on any one of the 16 powerpoints. Even if X places his first peg on a plane involving A there will be other planes involving A where O can place his second peg B; and on any such plane there will be 3 or 7 available powerpoints.

Even if X places his second peg on a plane including A & B there still remains a plane where O can place a third peg C onto one of the 2 or 6 available powerpoints. Here X must be foolish and allow a fourth O ‘1’ to be placed in that plane.

Once A, B, C & 1 are placed then there is a forced win with a further 5 pegs by O since X must respond with x1; then 2, x2; 3, x3 and so on.

.A.x3..3..B
.1..5..2..w
x1.x2..4...
.C....x4..w

Computer implementations

Several computer programs that play Qubic against a human opponent have been written.

William Daly Jr. wrote and described a Qubic-playing program as part of his Master's program at the Massachusetts Institute of Technology. The program was written in assembler language for the TX-0 computer. It included lookahead to 12 moves and kept a history of previous games with each opponent, modifying its strategy according to their past behavior.

An implementation in Fortran was written by Robert K. Louden and presented, with extensive description of its operation, in his book Programming the IBM 1130 and 1800. Its strategy involved looking for combinations of one or two free cells shared among two or three rows with particular contents.

A Qubic program in a DEC dialect of BASIC appeared in 101 BASIC Computer Games by David H. Ahi. Ahi said the program "showed up," author unknown, on a G.E. timesharing system in 1968.

A plotter based 3D computer game was written by Arthur Hu and Carl Hu in 1975 on a HP 9830 in Lindbergh High School. It used four stacked trapezoids. It was later ported to the HP 2647 demo tape with a graphical interface, using a simple mathematical transform to solve for 3D input position.

Three-dimensional tic-tac-toe on a 4x4x4 board was included in the Microsoft Windows Entertainment Pack in the 1990s as part of TicTactics.

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