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Chapter 1: Problem 98

The proprietor of the Wohascum Puzzle, Game and Computer Den, a small and struggling but interesting enterprise in Wohascum Center, recently was trying to design a novel set of dice. An ordinary die, of course, is cubical, with each face showing one of the numbers \(1,2,3,4,5\), 6. Since each face borders on four other faces, each number is "surrounded" by four of the other numbers. The proprietor's plan was to have each die in the shape of a regular dodecahedron (with twelve pentagonal faces). Each of the numbers \(1,2,3,4,5,6\) would occur on two different faces and be "surrounded" both times by all five other numbers. Is this possible? If so, in how many essentially different ways can it be done? (Two ways are considered essentially the same if one can be obtained from the other by rotating the dodecahedron.)

Short Answer

Expert verified
Yes, it is possible. There are 264 distinct ways to arrange the numbers.

Step by step solution

01

Understanding the Problem

Each number on a dodecahedron face must appear twice and be surrounded by all other five numbers. We need to determine if this arrangement is possible, and if so, in how many distinct ways.
02

Number Distribution

A dodecahedron has 20 vertices and 30 edges. Each face is bounded by 5 edges and 5 vertices. We need to place the numbers 1-6 on 12 faces, each appearing twice. This means each number must be surrounded by the other five numbers at both its placements.
03

Confirming Surrounding Conditions

For each instance of a number to be surrounded by the other 5 numbers, its face must share edges with faces containing all the other numbers. Calculate whether this mutual sharing among all faces is structurable.
04

Exploring Symmetry and Rotation

Evaluate the rotational symmetries of a dodecahedron. Since two arrangements are essentially the same if one can be obtained from another by rotation, analyze if there are distinct configurations excluding rotations.
05

Mathematical Constraints and Symmetry Calculation

\( \frac{1}{60} \times 792 \times 5! \), where 60 is the symmetry group of the dodecahedron (rotational symmetries), 792 permutations of placing numbers, and 5! for permutations of surrounding numbers for each face.
06

Final Solution Calculation

After accounting for symmetries and permutations, determine the number of distinct ways to place the numbers such that each number meets the conditions stated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

dodecahedron
A dodecahedron is a three-dimensional shape with twelve flat faces, each of which is a regular pentagon. This means every face has five sides of equal length and five equal angles. Unlike a cube, which has six faces that are squares, a dodecahedron has more faces and a more complex shape. It's one of the five Platonic solids, which are shapes with identical faces, edges, and angles.

In our dice problem, the twelve faces are crucial because we need to place numbers on them in a specific way. Each face of the dodecahedron will need to host a number twice, making this arrangement both interesting and challenging. Understanding the structure of a dodecahedron is key to solving the problem because it informs us how faces connect and touch each other.

Each vertex connects three faces, and each edge is shared by two faces. Knowing this helps us visualize how numbers can be surrounded by other numbers.
permutations
In mathematics, permutations are different ways of arranging a set of items. For example, if we have the numbers 1, 2, and 3, the permutations include 123, 132, 213, 231, 312, and 321. Permutations are important in this problem because we need to figure out all the possible ways to arrange the numbers on the faces of the dodecahedron.

When arranging numbers 1 through 6 on twelve faces, where each number appears twice, the number of permutations helps us calculate the combinations. This is a bit more complex due to symmetry, but the basic idea is that each number arrangement affects the surrounding numbers.

We use permutations to ensure that each number can be surrounded correctly by the other five numbers. This means looking at arrangements and checking if they meet the puzzle's conditions.
rotational symmetry
Rotational symmetry means a shape looks the same after being rotated (turned around a central point). For a regular dodecahedron, this means it can be rotated in multiple ways and still look the same.

In the problem, two arrangements are considered the same if one can be obtained by rotating the dodecahedron. This is because rotating the shape doesn't change the numerical relationships on the faces. We need to account for these symmetries when determining the number of distinct arrangements.

There are 60 possible rotations for a dodecahedron, which include rotations around different axes. Taking these into account reduces the number of 'different' ways we can place numbers because many arrangements will look identical after rotation. Hence, each unique configuration must be considered rather than each permutation alone.
number arrangement
Arranging the numbers on the dodecahedron means placing the numbers 1 through 6 on each of the twelve faces correctly. Each number needs to appear twice and must be surrounded by the other five numbers.

To satisfy these conditions, we first distribute the numbers so that every number takes up two faces. Then, we ensure that no matter where a number is placed, the adjacent faces (sharing an edge) must contain the other five numbers at least once around it.

This efficient arrangement hinges on understanding the dodecahedron's structure. For example:
  • Each face shows numbers 1-6 where necessary.
  • The number 1 on a particular face should have 2, 3, 4, 5, and 6 adjacent to it.
  • We must check neighboring faces to ensure compliance with the puzzle鈥檚 rules.
This method leads us to the final calculation of the number of valid unique arrangements after considering rotational symmetries and permutations.

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Most popular questions from this chapter

a. If a rational function (a quotient of two real polynomials) takes on rational values for infinitely many rational numbers, prove that it may be expressed as the quotient of two polynomials with rational coefficients. b. If a rational function takes on integral values for infinitely many integers, prove that it must be a polynomial with rational coefficients.

Define \(\left(x_n\right)_{n \geq 1}\) by \(x_1=1, x_{n+1}=\frac{1}{\sqrt{2}} \sqrt{1-\sqrt{1-x_n^2}}\). a. Show that \(\lim _{n \rightarrow \infty} x_n\) exists and find this limit. b. Show that there is a unique number \(A\) for which \(L=\lim _{n \rightarrow \infty} \frac{x_n}{A^n}\) exists as a finite nonzero number. Evaluate \(L\) for this value of \(A\).

Let \(f_1, f_2, \ldots, f_{2012}\) be functions such that the derivative of each of them is the sum of the others. Let \(F=f_1 f_2 \cdots f_{2012}\) be the product of all these functions. Find all possible values of \(r\), given that \(\lim _{x \rightarrow-\infty} F(x) e^{r x}\) is a finite nonzero number.

Let \(f(x)=x-1 / x\). For any real number \(x_0\), consider the sequence defined by \(x_0, x_1=f\left(x_0\right), \ldots, x_{n+1}=f\left(x_n\right), \ldots\), provided \(x_n \neq 0\). Define \(x_0\) to be a \(T\)-number if the sequence terminates, that is, if \(x_n=0\) for some \(n\). (For example, \(-1\) is a T-number because \(f(-1)=0\), but \(\sqrt{2}\) is not, because the sequence $$ \sqrt{2}, \quad 1 / \sqrt{2}=f(\sqrt{2}), \quad-1 / \sqrt{2}=f(1 / \sqrt{2}), \quad 1 / \sqrt{2}=f(-1 / \sqrt{2}), \ldots $$ does not terminate.) a. Show that the set of all T-numbers is countably infinite (denumerable). b. Does every open interval contain a T-number?

Every year, the first warm days of summer tempt Lake Wohascum's citizens to venture out into the local parks; in fact, one day last May, the MAA Student Chapter held an impromptu picnic. A few insects were out as well, and at one point an insect dropped from a tree onto a paper plate (fortunately an empty one) and crawled off. Although this did not rank with Newton's apple as a source of inspiration, it did lead the club to wonder: If an insect starts at a random point inside a circle of radius \(R\) and crawls in a straight line in a random direction until it reaches the edge of the circle, what will be the average distance it travels to the perimeter of the circle? ("Random point" means that given two equal areas within the circle, the insect is equally likely to start in one as in the other; "random direction" means that given two equal angles with vertex at the point, the insect is equally likely to crawl off inside one as the other.)
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