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Chapter 16: Problem 1

In how many ways can we 5-color the vertices of a square that is free to move in (a) two dimensions? (b) three dimensions?

Short Answer

Expert verified
The answer is (a) \(5^4/4\) ways in 2 dimensions and (b) \(5^4/(4*2)\) ways in 3 dimensions.

Step by step solution

01

Count 5-colorings in 2D view

In a flat, 2-dimensional view, there is no difference in coloring whether it is in top-down view or flipped. So, the method here is to calculate the number of 5 different colorscombinaions for 4 vertices of the square. Since each vertex can be any of 5 colors, there are \(5^4\) possibilities.
02

Remove redundant count for rotation in 2D

In the 2D case, the square can rotate 90 degrees, 180 degrees and 270 degrees. So the same color configuration for a square is over-counted 4 times (including the original position). Therefore, we divide our original \(5^4\) by 4 to get the true distinct colorings considering rotation, yielding \(5^4/4\) possibilities.
03

Count 5-colorings in 3D view

In a 3-dimensional view, the square can be flipped around its center to reveal the other side. All colorings have a mirror image due to flipping, so the situations are just like in the 2D view, with one difference: because of flipping, each coloring is counted twice. This leads to \(5^4/2\).
04

Remove redundant count for rotation in 3D

In the 3D case, similar to 2D case, one has to consider the rotation which can generate repetitions in the count. As a result, each coloring is over-counted 4 times due to the 90, 180 and 270 degrees rotations and the initial state. Thus dividing by 4 leads to the answer of \(5^4/(4*2)\).

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Most popular questions from this chapter

a) In how many ways can we 5 -color the vertices of a regular hexagon that is free to move in two dimensions? b) Answer part (a) if the hexagon is free to move in three dimensions. c) Find two 5 -colorings that are equivalent for case (b) but distinct for case (a).

A baton is painted with three cylindrical bands of color (not necessarily distinct), with each band of the same length. a) How many distinct paintings can be made if there are three colors of paint available? How many for four colors? b) Answer part (a) for batons with four cylindrical bands. c) Answer part (a) for batons with \(n\) cylindrical bands. d) Answer parts (a) and (b) if adjacent cylindrical bands are to have different colors.

Find the elements in the groups \(U_{20}\) and \(U_{24}\)-the groups of units for the rings \(\left(\mathbf{Z}_{20},+, \cdot\right)\) and \(\left(\mathbf{Z}_{24},+, \cdot\right)\), respectively.

Let \(\omega\) be the complex number \((1 / \sqrt{2})(1+i)\). a) Show that \(\omega^{8}=1\) but \(\omega^{n} \neq 1\) for \(n \in \mathbf{Z}^{+}, 1 \leq n \leq 7 .\) b) Verify that \(\left\\{\omega^{n} \mid n \in \mathbf{Z}^{*}, 1 \leq n \leq 8\right\\}\) is an abelian group under multiplication.

If a regular pentagon is free to move in space and we can color its vertices with red, white, and blue paint, how many nonequivalent configurations have exactly three red vertices? How many have two red, one white, and two blue vertices?
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