Question

You have two colours of paint. In how many different ways can you paint the faces of a cube if each face is painted? Painted cubes are considered to be the same if you can rotate one cube so that it matches the other one exactly.

Short answer

10

Step-by-step solution

- Understand Cube Symmetry

First, recognize that a cube has 6 faces, and due to its symmetrical properties, different rotations of the cube can produce identical arrangements. There are 24 unique rotations for a cube.

- Identify Colour Combinations

You need to consider two colors, say Color 1 and Color 2. We need to count how many unique ways we can distribute these two colors across the 6 faces with respect to cube rotations.

- Calculate Cases with Same Colors

There are extreme cases where all faces are the same color. If all faces are Color 1, this results in only 1 combination. Likewise, if all faces are Color 2, this results in 1 combination. Thus, we have 2 combinations from these two cases.

- Calculate Cases with One Face Different

Consider configurations with 5 faces painted one color and 1 face painted the other color. There are 6 faces, so we can choose any one of them to be different. Thus, we have 6 combinations.

- Calculate Cases with Two Faces Different

If two faces are one color and the remaining four another color, due to cube symmetry, only 1 unique arrangement is possible (two opposite or adjacent faces differently colored will be equivalent under some rotation). Thus, we have 1 combination.

- Calculate Cases with Three Faces Different

If three faces are one color and the other three are the other color, only one unique arrangement can be achieved by symmetries (alternating or adjacent faces divided equally). Thus, we have 1 combination.

- Summarize Unique Combinations

Add up all the unique combinations found in the previous steps: 2 (all same) + 6 (one different) + 1 (two different) + 1 (three different). This gives a total of 10 unique painted cubes.

Key concepts

cube symmetry

When examining cube symmetry, it's important to recognize that a cube has six faces. Due to its symmetrical properties, different rotations of the same arrangement can look identical. This means we need to factor in not only the painting pattern but also how the cube can be turned.
Every cube has 24 unique rotations, which represent all the ways a cube can be turned in space. When painting the cube with different colors, each arrangement needs to be considered up to these rotations to determine if it’s truly unique. The concept is that two painted cubes are deemed identical if one can be rotated to match the other perfectly.
Understanding cube symmetry helps us rule out many configurations that might initially seem different but are actually identical when rotated. This reduces our calculation workload significantly.

color combinations

Given two colors and six faces to paint, there are multiple ways to distribute these colors. For simplicity, let's name our colors Color 1 and Color 2. We need to account for all possible color combinations.
Here are a few key cases to consider:

• If all faces are painted the same color, there are only 2 possibilities (all faces Color 1 or all faces Color 2).
• If one face is different from the other five, there are 6 ways to paint one face a different color since we can choose any one out of the six faces.
• If two faces are of one color and the remaining four are the other color, the specific arrangement of these two differently colored faces plays a role.
• If three faces are one color and the remaining three are the other color, this results in a balanced distribution. Due to symmetry, only one unique arrangement is possible.

By systematically handling each color combination scenario, we can better manage the counting process.

rotational symmetry

Rotational symmetry in cubes plays a crucial role in counting unique color arrangements. Since a cube looks the same from multiple angles due to its 24 rotational symmetries, many painted arrangements can look identical if rotated. This concept helps eliminate redundant calculations.
Let’s break down what rotational symmetry means in this context:

• If all faces are the same color, rotations do not create any new distinctions. It’s the same cube regardless of rotation.
• If one face is different, rotating the cube keeps the unique arrangement as long as the position of the distinct face is considered.
• If two faces are different, we must consider if these faces are opposite or adjacent in order to determine if rotations can make them appear identical.
• For three different faces, whether these faces are grouped or spaced apart and the cube’s symmetry can make multiple different rotations appear identical.

By understanding and applying rotational symmetry, we simplify the problem significantly.

counting unique arrangements

To find the number of unique painted cubes, we start by breaking down and counting combinations. Summarizing our findings within the rules of cube symmetry and color distribution:

• All faces the same color: 2 unique combinations (all Color 1 or all Color 2).
• One face different: 6 combinations (one face Color 1, five faces Color 2 and vice versa).
• Two faces different: 1 unique combination since any two differently painted faces will appear the same under rotation.
• Three faces different: 1 unique combination as rotational symmetry will make all such configurations identical.

Adding these gives us:
2 (all same) + 6 (one different) + 1 (two different) + 1 (three different) = 10 unique combinations.
This careful approach ensures that all rotations and symmetries are accounted for, giving us the correct count of unique painted cube arrangements.