We will first prove a lemma that all connected groups of green blocks that are completely surrounded by red or blue blocks will produce an even number of trichromatic triangles. We will then augment the triangle by adding an extra blue row on the bottom and an extra red side on the right such that we now have would what be a triangle if it weren’t missing the bottom right corner. This will mean that all green blocks will now be surrounded so we’ll have an even number of trichromatic triangles, but we have added exactly one additional such triangle, so that the original has an odd number.
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Proof of Lemma:
A trivial variant of 1-d Sperner’s Lemma is that if we start and finish on the same colour, we get an even number of bichromatic edges. For any block that is completely surrounded by red and blue blocks, we apply this variant to show that there is an even number of bichromatic edges on the outside that then translates to an even number of trichromatic triangles.
EDIT: Actually, there is one case we can run into which is tricky and that is something like:
R R R R R
R G G G R
R G R B R
R G G G R
R R R R R R
To see how to solve this case, read how to solve tricky interior cases below.
END EDIT
Thick interior blocks work similarly, but we can run into weird scenarios such as a blue and a red surrounding by green block:
g g g
g b r g
g g g
We might also run into something like this (ie. a three-spoked shape that is blue at the center and red on the sides surrounded by green).
g g g
g r g g
g b r g
g r g g
g g g
For these weird shapes we can still trace a path around this interior section, we just include going both down and up a spoke in our path (thanks Hoagy!). So the interior sections are also even.
Here’s a solution to 4:
We will first prove a lemma that all connected groups of green blocks that are completely surrounded by red or blue blocks will produce an even number of trichromatic triangles. We will then augment the triangle by adding an extra blue row on the bottom and an extra red side on the right such that we now have would what be a triangle if it weren’t missing the bottom right corner. This will mean that all green blocks will now be surrounded so we’ll have an even number of trichromatic triangles, but we have added exactly one additional such triangle, so that the original has an odd number.
------
Proof of Lemma:
A trivial variant of 1-d Sperner’s Lemma is that if we start and finish on the same colour, we get an even number of bichromatic edges. For any block that is completely surrounded by red and blue blocks, we apply this variant to show that there is an even number of bichromatic edges on the outside that then translates to an even number of trichromatic triangles.
EDIT: Actually, there is one case we can run into which is tricky and that is something like:
R R R R R
R G G G R
R G R B R
R G G G R
R R R R R R
To see how to solve this case, read how to solve tricky interior cases below.
END EDIT
Thick interior blocks work similarly, but we can run into weird scenarios such as a blue and a red surrounding by green block:
g g g
g b r g
g g g
We might also run into something like this (ie. a three-spoked shape that is blue at the center and red on the sides surrounded by green).
g g g
g r g g
g b r g
g r g g
g g g
For these weird shapes we can still trace a path around this interior section, we just include going both down and up a spoke in our path (thanks Hoagy!). So the interior sections are also even.