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. | + | 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. | + | 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 09:29, 24 July 2020
Problem
How many different ways are there to paint the sides of a tetrahedron with exactly 
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 colors - 
and 
Fix the bottom as color 
On the remaining sides you can have colors 
(in that order), or 
which are not rotationally identical. So, there are 
ways to color it.
Credit
Problem and solution were taken from https://sumo.stanford.edu/old/smt/2000/
