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Chapter 1: Problem 206

Define a die to be a convex polyhedron. For what \(n\) is there a fair die with \(n\) faces? By fair, we mean that, given any two faces, there exists a symmetry of the polyhedron which takes the first face to the second.

Short Answer

Expert verified
A fair die can only have 4, 6, 8, 12, or 20 faces.

Step by step solution

01

- Understand the definition of a fair die

A fair die is a convex polyhedron where all faces are geometrically identical, and any rotation can map one face to any other face.
02

- Identify symmetric polyhedra

Examine known regular polyhedra (Platonic solids). These are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, which have 4, 6, 8, 12, and 20 faces, respectively.
03

- Confirm symmetry properties

Each Platonic solid is highly symmetric, meaning every face is the same in shape and size. Additionally, there is a symmetry operation (rotation) that can map any given face to any other face.
04

- Determine possible values for n

Since the regular (Platonic) polyhedra are the only convex polyhedra that have this level of symmetry, the possible values for n are 4, 6, 8, 12, and 20.
05

- Conclude with valid n values

Thus, a fair die can only have 4, 6, 8, 12, or 20 faces.

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Key Concepts

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

Convex Polyhedron
A convex polyhedron is a three-dimensional solid where all interior angles between faces are less than 180 degrees. This ensures the polyhedron doesn't cave in on itself. All vertices (corners) point outwards. To visualize, think of a classic dice or a pyramid. Here are some key points:
  • Each face of a convex polyhedron is a flat polygon.
  • The overall shape bulges outward, with no indentations.
  • Examples include cubes, octahedrons, and dodecahedrons.
Understanding convex polyhedra helps in grasping more complex geometrical concepts like symmetry and regular solids.
Platonic Solids
Platonic solids are a special category of convex polyhedra with highly symmetric properties. They are named after the ancient Greek philosopher Plato who studied them. Each Platonic solid is defined by:
  • All faces are identical regular polygons.
  • The same number of faces meet at each vertex.
  • Examples: Tetrahedron (4 faces), Cube (6 faces), Octahedron (8 faces), Dodecahedron (12 faces), and Icosahedron (20 faces).
These solids are the only convex polyhedra that can make fair dice because their symmetry allows any face to be mapped to any other face by rotating the solid. This unbiased face distribution is essential for fairness in dice.
Geometric Symmetry
Geometric symmetry means that a shape looks the same after some transformations, such as rotations or reflections. This concept is crucial in defining a fair die:
  • Each face of a fair die must be able to align with any other face through a rotation.
  • This level of symmetry ensures that no face is uniquely disadvantaged or favored.
  • Common symmetry operations include rotations around the center and reflections across planes.
Understanding geometric symmetry is key to visualizing how Platonic solids can uniformly randomize face landing in a dice roll. This uniform randomization ensures that all faces have an equal chance of landing face up.

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Most popular questions from this chapter

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 \(A\).

Suppose we are given an \(m\)-gon (polygon with \(m\) sides, and including the interior for our purposes) and an \(n\)-gon in the plane. Consider their intersection; assume this intersection is itself a polygon (other possibilities would include the intersection being empty or consisting of a line segment). a. If the \(m\)-gon and the \(n\)-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the polygons is assumed to be convex? (Note: A subset of the plane is convex if for every two points of the subset, every point of the line segment between them is also in the subset. In particular, a polygon is convex if each of its interior angles is less than \(\left.180^{\circ}.\right)\)

Suppose you draw \(n\) parabolas in the plane. What is the largest number of (connected) regions that the plane may be divided into by those parabolas? (The parabolas can be positioned in any way; in particular, their axes need not be parallel to either the \(x\) - or the \(y\)-axis.)

Let \(A \neq 0\) and \(B_1, B_2, B_3, B_4\) be \(2 \times 2\) matrices (with real entries) such that $$ \operatorname{det}\left(A+B_i\right)=\operatorname{det} A+\operatorname{det} B_i \quad \text { for } i=1,2,3,4 . $$ Show that there exist real numbers \(k_1, k_2, k_3, k_4\), not all zero, such that $$ k_1 B_1+k_2 B_2+k_3 B_3+k_4 B_4=0 . $$ ( 0 is the zero matrix, all of whose entries are 0 .)

Let \(a\) and \(b\) be positive real numbers, and define a sequence \(\left(x_n\right)\) by $$ x_0=a, x_1=b, x_{n+1}=\frac{1}{2}\left(\frac{1}{x_n}+x_{n-1}\right) . $$ a. For what values of \(a\) and \(b\) will this sequence be periodic? b. Show that given \(a\), there exists a unique \(b\) for which the sequence converges. (p. 357\()\)
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