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Can you avoid monochromatic triangles? Question
In the diagram below we have pentagon ABCDE with 5 edges and 5 diagonals. These 10 line segments are currently all colored black.
The diagram contains several triangles but for this puzzle we are only interested in the 10 triangles where all three of its vertices are vertices of pentagon ABCDE.
A triangle is considered monochromatic if all three of its edges are the same color. Currently all 10 triangles are monochromatic black.
Is it possible to recolor some of the 10 line segments red, leaving the other line segments black, so that none of the 10 triangles are monochromatic?
If yes, provide a diagram showing the recoloring.
If no, prove it is impossible.
2 answers
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| User | Comment | Date |
|---|---|---|
| will.octagon.gibson | (no comment) | Aug 1, 2026 at 21:28 |
I'm not sure if I'm doing this right, but I have a pentagram that passes my test.
#! /usr/bin/env python
AB = 1
AC = 0
AD = 1
AE = 0
BC = 0
BD = 0
BE = 1
CD = 1
CE = 1
DE = 0
triangles = {
# Obtuse triangles, clockwise from top
'ABE': AB + BE + AE, # ⏶
'ABC': AB + BC + AC, # ◥
'BCD': BC + CD + BD, # ◢
'CDE': CD + DE + CE, # ◣
'ADE': AD + DE + AE, # ◤
# Acute triangles, clockwise from top
'ACD': AC + CD + AD, # △
'BDE': BD + DE + BE, # ◹
'ACE': AC + CE + AE, # ◿
'ABD': AB + BD + AD, # ◺
'BCE': BC + CE + BE, # ◸
}
print({t: cs for t, cs in triangles.items() if cs in {0, 3}})
Output
{}
Edit: Now that I ponder this for a moment longer, the easiest way is to make the exterior pentagon one color and all interior lines a different color. Every triangle will have at least one face on the exterior, but not all three.
0 comment threads
Yes.
One answer is the five exterior line segments one color and the five interior line segments another.
The reasoning is that all 10 of the triangles you care about have both exterior and interior segments. Therefore, with exterior and interior segments different colors, all of the 10 triangles must be not monochromatic.
There are possibly other solutions. I didn't look into that.
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