Travis.russell: Combinations of the Tower Cube (HW1-Travis Russell)

2014-10-05T23:27:02Z

Combinations of the Tower Cube (HW1-Travis Russell)

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== 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:

[[File:Tower_cube.png]]

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

[[File:Tower_cube_generators.png]]

Where <math>g_1</math> is the generator that fixes the yellow face and makes a quarter-turn about the blue face, <math>g_2</math> is the generator that makes a quarter-turn about the middle tier of blocks, <math>g_3</math> is the generator that fixes the blue face and makes a quarter-turn about the yellow face, <math>g_4</math> is the generator that fixes the red face and makes a half-turn about the blue face, <math>g_5</math> is the generator that fixes the orange face and makes a half turn about the red face, <math>g_6</math> is the generator that fixes the green face and makes a half-turn about the white face, and <math>g_7</math> 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:

[[File:Tower_cube_program.png]]

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:

[[File:Tower_cube_output.png]]

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 [http://www.jaapsch.net/puzzles/cube223.htm 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.

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Travis.russell: Combinations of the Tower Cube (HW1-Travis Russell)