Ernö Rubik, a lecturer at the Academy of Applied Arts and Design in Budapest, Hungary, made the prototype of his famous cube in 1974, as an exercise in design and structural problem solving. In the process, he created a puzzle of almost limitless possibilities. This array of small cubes, known as cubies, can be twisted and turned into 43 quintillion different configurations.
Fresh out of the package, all nine cubie faces on each side of a Rubik’s Cube are the same color, as shown above left. An internal mechanism lets you twist any 3 x 3 slice of cubies by 90 degrees. After just a few twists, the colors become hopelessly mixed up. The challenge is to twist the scrambled cubies back to their original positions.
1. [Challenging] Imagine that Rubik’s Cube is composed of 27 (3 x 3 x 3) cubies. (In an actual Rubik’s Cube, the cubies are not full cubes, and there’s no center cube.) Only the outside faces of the cube are colored; all inside faces are black. How many of the cubies are colored on one side? On two sides? On three sides? Four? Five? Six? No sides?
2. [Easy] Which pairs of colors never appear together on the same cubie?
3. [Easy] How many different ways are there to twist a face of Rubik’s Cube by 90 degrees?
4. [Challenging] When you twist one face, which of the 27 cubies stay in the same position?
5. [Difficult] Suppose you are trying to twist a mixed-up cube back to its original position. How can you tell which face should be which color?