§ 1D

• Given a line with endpoints $a, b$ and points in between, we will always have an occurrence of $ab$ on the line.
• Can prove something slightly stronger: there will always be an odd number of $ab$ on the line.

§ 2D

• Given a triangle labelled $abc$ and a subdivision of it, there will be a smaller triangle labelled $abc$.
• Consider all smaller triangles.
• Call a side with $bc$ a door.
• How many doors can a triangle have? It can have 0 doors if it is labelled $aaa$, or $abb$, or some such.
• It can have 1 door if it is:

 a
/ \
b==c

• It can have two doors if it is:

  c
// \
b===c

• We can't have three doors. So triangles can have 0, 1, or 2 doors.
• If we find a triangle with one door, we are done, since it will have $abc$.
• Now start from the bottom of the triangle where we have the side $bc$. Here we will find at least one edge $bc$.
• Walk along the triangle, entering any triangle with a door.
• If that's the only door of the triangle, we are done.
• If not, then the triangle has two doors. Exit the current triangle through the other door (the door we did not enter from). This will take us to another triangle.
• See that we cannot terminate the walk by exiting from sides $AB$ or $AC$, for such a side will be of the form $ab$. Then to have a door, we will get a $bc$, so the triangle must be $abc$, ie a triangle we are looking for!
• So if we ever escape the simplex, we must escape from the bottom side $BC$. This removes an even number of $bc$ edges from $BC$. But we know there are an odd number of $bc$ from $BC$, so we must find a triangle $ABC$ eventually.