Difference between revisions of "2000 SMT/Advanced Topics Problems/Problem 1"

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==Problem==
 
==Problem==
  
How many different ways are there to paint the sides of a tetrahedron with exactly 4 colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.
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How many different ways are there to paint the sides of a tetrahedron with exactly <math>4</math> colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.
  
  
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==SMT Solution==
 
==SMT Solution==
  
Assume we have 4 colors - 1, 2, 3, and 4. Fix the bottom as color 1. On the remaining sides you can have colors 2, 3, 4 (in that order), or 2, 4, 3, which are not rotationally identical. So, there are <math>\mathbf{2}</math> ways to color it.
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Assume we have <math>4</math> colors - <math>1, 2, 3,</math> and <math>4.</math> Fix the bottom as color <math>1.</math> On the remaining sides you can have colors <math>2, 3, 4</math> (in that order), or <math>2, 4, 3,</math> which are not rotationally identical. So, there are <math>\mathbf{2}</math> ways to color it.
  
  

Latest revision as of 08:29, 24 July 2020

Problem

How many different ways are there to paint the sides of a tetrahedron with exactly $4$ colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.


SMT Solution

Assume we have $4$ colors - $1, 2, 3,$ and $4.$ Fix the bottom as color $1.$ On the remaining sides you can have colors $2, 3, 4$ (in that order), or $2, 4, 3,$ which are not rotationally identical. So, there are $\mathbf{2}$ ways to color it.




Credit

Problem and solution were taken from https://sumo.stanford.edu/old/smt/2000/