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Chapter 16: Problem 10

A pyramid has a square base and four faces that are equilateral triangles. If we can move the pyramid about (in three dimensions), how many nonequivalent ways are there to paint its five faces if we have paint of four different colors? How many if the color of the base must be different from the color(s) of the triangular faces?

Short Answer

Expert verified
With no restrictions, there are 1024 nonequivalent ways to paint the pyramid's five faces. If the base must be a different colour than the triangle faces, there are 12 nonequivalent ways to paint the pyramid.

Step by step solution

01

Figuring out the number of ways to paint the pyramid with no restrictions

With no restrictions on the colours, each face of the pyramid (including the base) can be painted in 4 different ways regardless of the colour of the other faces. Therefore, since there are 5 faces (1 square and 4 triangles), we can use the counting principle to say that there are \(4^5\) ways to paint the five faces of the pyramid.
02

Calculating the total number of ways

Calculating \(4^5 = 1024\), there are a total of 1024 ways to paint the pyramid with no restrictions.
03

Figuring out the number of ways to paint the pyramid with restrictions

The second part of the problem implies a restriction: the base and the triangle faces cannot be the same colour. First, paint the base, it can be painted in any of the four colours (4 options exist). The four triangular faces then can be painted in any of the remaining three colours. As these faces are indistinguishable from each other when rotating the pyramid in three dimensions, they are considered to be identical. So, there are only 3 options for painting these faces.
04

Calculating the total number of ways with restrictions

Using the counting principle again, the total number of ways the pyramid can be painted, given the restrictions, is 4 (options for the base) times 3 (options for the triangles), or \(4 * 3 = 12\). Therefore, there are 12 ways to paint the pyramid given the restrictions.

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Most popular questions from this chapter

For each of the following sets, determine whether or not the set is a group under the stated binary operation. If so, determine its identity and the inverse of each of its elements. If it is not a group, state the condition(s) of the definition that it violates. a) \(\\{-1,1\\}\) under multiplication b) \(\\{-1,1\\}\) under addition c) \(\\{-1,0,1\\}\) under addition d) \(\\{10 n \mid n \in \mathbf{Z}\\}\) under addition e) The set of all functions \(f: A \rightarrow A\), where \(A=\\{1,2,3,4\\}\), under function composition f) The set of all one-to-one functions \(g: A \rightarrow A\), where \(A=\\{1,2,3,4\\}\), under function composition g) \(\left\\{a / 2^{n} \mid a, n \in \mathbf{Z}, n \geq 0\right\\}\) under addition

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