14-1100/tower cube

The Tower Cube - Homework 1, Travis Russell

The Tower Cube is a combination puzzle reminiscent of the Rubik's Cube. While the normal Rubik's Cube is a 3x3x3 cube, the Tower Cube is 2x2x3, giving it a distinct tower shape. The 2x2 faces of the Tower Cube are able to make 90 degree turns, (or quarter-turns), while the 2x3 faces of the Tower Cube can only make 180 degree turns (or half-turns). Here is a diagram of the Tower Cube with its vertices labelled:

In order to generate the possible combinations of the Tower Cube, I used the following seven generators:

Where $g_{1}$ is the generator that fixes the yellow face and makes a quarter-turn about the blue face, $g_{2}$ is the generator that makes a quarter-turn about the middle tier of blocks, $g_{3}$ is the generator that fixes the blue face and makes a quarter-turn about the yellow face, $g_{4}$ is the generator that fixes the red face and makes a half-turn about the blue face, $g_{5}$ is the generator that fixes the orange face and makes a half turn about the red face, $g_{6}$ is the generator that fixes the green face and makes a half-turn about the white face, and $g_{7}$ is the generator that fixes the white face and makes a half-turn about the green face.

In order to compute the number of possible combinations of the Tower Cube, I used the code given in class by Professor Bar-Natan:

I attempted to write my own program using Java, but I encountered some stack overflow errors. Perhaps in the future I will update this page with my own solution.

After running the program, the following output was generated:

Analyzing the output, it seems apparent that only one of the half-turn generators is necessary; combining the three quarter-turn generators with one of the half-turn generators will give all possible combinations. Checking our solution with other sources online, (such as this one), we see that the correct solution is 241920, or exactly one eighth of the combinations we generated. In the classic version of the puzzle, the 8 corner pieces are not oriented, however using our generators they are, and this accounts for the factor of eight in the difference between solutions.