• ## Playing with model trains and calling it graph theory

You’ve probably played with model trains, for instance with something like the Brio set shown below.1 And if you’ve built a layout with a model train set, you may well have wondered: is it possible for my train to use all the parts of my track?

1. Searching on tineye finds that this image was on Amazon in 2008. Presumably it was supplied to them by Brio?

• ## Euler characteristics of non-manifold polycubes

From a $2\times 2$ block of cubes, remove two non-adjacent and non-opposite cubes. The resulting polycube has a boundary that is not a manifold: between the two removed cubes, there is an edge shared by four squares, but a two-dimensional manifold can only have two faces per edge. Nevertheless, we can compute its Euler characteristic as the number of vertices ($25$) minus the number of edges ($47$) plus the number of square faces ($24$). $25-47+24=2$, the same number we would expect for the Euler characteristic of a topological sphere! What does it mean?

• ## Monochromatic grids in colored grids

Color the points of an $n\times n$ grid with two colors. How big a monochromatic grid-like subset can you find? By “grid-like” I mean that it should be possible to place equally many horizontal and vertical lines, partitioning the plane into $k\times k$ cells each of which contains a single point.

• ## Coloring kinggraphs

Draw a collection of quadrilaterals in the plane, meeting edge to edge, so that they don’t surround any open space (the region they cover is a topological disk) and every vertex interior to the disk touches at least four quadrilaterals. Is it always possible to color the corners of the quadrilaterals with four colors so that all four colors appear in each quadrilateral?