Problem 5 - Algebra: Pure and Applied

Question

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Color the six faces of a cube with six different colors, if seven colors are available and no color is to be used more than once.

Step-by-Step Solution

Answer

There are 210 essentially different ways to color the cube.

Step 1: Understand the Problem

We are tasked with finding the number of different ways to color a cube given six faces and seven available colors. Each face must have a different color. Importantly, this is a geometric problem where the orientation of the cube affects the coloring.

Step 2: Calculate Total Permutations of Colors

First, choose 6 out of the available 7 colors to use on the cube. This can be done in \( \binom{7}{6} = 7 \) ways. Then, permute these 6 colors to color the cube, which can be done in \(6! = 720\) ways.

Step 3: Account for Cube Symmetries

There are 24 different orientations in which a cube can be rotated without changing its essential appearance. To get the number of distinct colorings, divide the total permutations by the number of these symmetries: \(\frac{7 \times 720}{24} = 210\).

Step 4: Interpret the Result

So, the number of essentially different ways to color the cube is 210. This accounts for each unique configuration after considering all rotations of the cube.

Key Concepts

Permutations

Permutations are arrangements of objects in a specific order. In this exercise, permutations allow us to determine how we can arrange colors on the cube's faces. When we choose 6 colors out of 7 and arrange them on 6 faces, each different "arrangement" is a permutation.

• The number of ways to choose 6 colors from 7 is given by the combination formula: \( \binom{7}{6} = 7 \).
• Once we've selected our 6 colors, we need to arrange them on the 6 faces. This is where permutations come in. With each color going on one face, we find the order using the formula for permutations of 6 distinct items: \(6! = 720\).

This means there are 720 possible ways to arrange our chosen 6 colors.

Symmetries

Symmetries in geometry refer to transformations that can be made to an object without altering the overall appearance. For a cube, these are the possible rotations that make it look the same.

• There are 24 unique orientations a cube can have. This is determined by considering how many ways we can rotate the cube about its center without changing the arrangement.
• Each rotation needs to be accounted for to ensure we don't double-count any color arrangements. This makes symmetries crucial in reducing the total permutations to only those that are "different" in essence, not just by rotation.

Thus, when calculating distinct cube colorings, we must divide the total permutations by the 24 cube symmetries.

Combinations

Combinations involve selecting items from a larger set where the order doesn't matter. They are crucial for the first step of our problem when we choose which colors to use.

• Given 7 colors, we need to select 6 to paint the cube. Here, the order of selection does not matter; we're only concerned with which colors will be used.
• This is calculated as \( \binom{7}{6} \), representing the number of ways to choose 6 colors from 7 without caring about order.

Combinations help us narrow down our choices before considering the order with permutations.

Geometric Problem

This exercise is fundamentally a geometric problem involving spatial reasoning and rotational understanding.

• We deal with a cube, a 3D shape, adding complexity since its positioning affects how we perceive its coloration.
• The challenge lies not just in arranging colors but understanding how these arrangements appear when viewed or rotated differently.
• As such, this problem ties algebraic operations (counting permutations and combinations) to geometric manipulations (accounting for symmetries).

By carefully considering both algebraic and geometric aspects, we find the true number of unique solutions.