Color the edges and diagonals of a regular polygon

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Here is the problem:



For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle?



Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks




combinatorial-geometry




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  • Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
    – hardmath
    Aug 6 at 21:20






  • 3




    $n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
    – Mike Earnest
    Aug 6 at 21:58






  • 1




    See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
    – Mike Earnest
    Aug 6 at 22:16














up vote
5
down vote

favorite
1












Here is the problem:



For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle?



Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks




combinatorial-geometry




share|cite|improve this question





















  • Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
    – hardmath
    Aug 6 at 21:20






  • 3




    $n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
    – Mike Earnest
    Aug 6 at 21:58






  • 1




    See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
    – Mike Earnest
    Aug 6 at 22:16












up vote
5
down vote

favorite
1









up vote
5
down vote

favorite
1






1





Here is the problem:



For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle?



Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks




combinatorial-geometry




share|cite|improve this question













Here is the problem:



For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle?



Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks





combinatorial-geometry





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 21:57









alcana

1184




1184









asked Aug 6 at 21:15









Leo Gardner

35611




35611











  • Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
    – hardmath
    Aug 6 at 21:20






  • 3




    $n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
    – Mike Earnest
    Aug 6 at 21:58






  • 1




    See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
    – Mike Earnest
    Aug 6 at 22:16
















  • Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
    – hardmath
    Aug 6 at 21:20






  • 3




    $n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
    – Mike Earnest
    Aug 6 at 21:58






  • 1




    See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
    – Mike Earnest
    Aug 6 at 22:16















Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
– hardmath
Aug 6 at 21:20




Even assuming "polyhpn" in the title is a typo for polygon, it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges.
– hardmath
Aug 6 at 21:20




3




3




$n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
– Mike Earnest
Aug 6 at 21:58




$n$ needs to be odd; focusing on a vertex, $v$, the $n-1$ edges out of $v$ are divided into same-colored pairs.
– Mike Earnest
Aug 6 at 21:58




1




1




See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
– Mike Earnest
Aug 6 at 22:16




See site.uottawa.ca/~lucia/courses/7160-17/slides/… for a construction
– Mike Earnest
Aug 6 at 22:16















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