# Bogglers

By Scott Kim|Sunday, February 6, 2005
Starting position
Front and inside
Twisting
Scrambled

Rubik's cubism
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.

The Cubies

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?

Challenging
Difficult
Very difficult

The Colors

The colors on a Rubik’s Cube are stickers that have been applied to the black plastic cubies. You can even cheat by pulling off the stickers and gluing them back on in the right places. (If you want to indulge in such self-trickery, the official Rubik’s site, www.rubiks.com, sells blank cubes and stickers.)

Each pair of diagrams below shows a mixed-up cube. The diagrams on the left show the front faces of a cube; the diagrams on the right show the back faces of the cube as if the front were transparent. The light gray faces are missing their stickers. Can you deduce the missing colors?

A
B

The Variations

Like all popular toys, Rubik’s Cube has given rise to a host of competitors. Puzzle inventor Uwe Meffert of Hong Kong (www.mefferts.com) has devised more than a hundred riffs on Rubik’s theme, including Pyraminx (A), which has four triangular faces, and the charmingly eccentric Skewb Cube (B).

As with Rubik’s Cube, you start out by twisting the initially solid-colored faces of Skewb Cube and Pyraminx to mix up the colors. The goal is then to unscramble the colors back to their original order.

1. [Easy] How many stickers of how many shapes and colors do you need to cover the surface of Pyraminx? And of Skewb Cube?

2. [Easy] Rubik’s Cube has six different faces that can be twisted around three perpendicular axes. How many different twists and how many axes are there for Pyraminx and the Skewb Cube? Hint: Each axis for Pyraminx is perpendicular to a face.

3. [Challenging] The faces of Rubik’s Cube are made up of three types of small cubies: 8 corner cubies colored on three sides, 12 edge cubies colored on two sides, and 6 center cubies colored on one side. How many types of pieces are in Pyraminx and the Skewb Cube, and how many of each type?