Coloring a donut

Let's say that we have a map which is split into many different regions. We want to color each region such that no two regions of the same color share a border. It has been proven that only four distinct colors are required, no matter what the map is.

There are two parts to the proof:

1. Some map requires at least four colors.
2. No map requires more than four colors.

Part 2 is complicated! It required computer assistance to check over a thousand cases.

But part 1 is easy. Here is one such map.

One of the assumptions of this theorem is that the map is on a flat plane. But what if we have a map on a donut's surface? It turns out that we need seven colors. Can you find a map on a donut that requires at least seven colors?

You can send solutions to skepticsplay at gmail dot com. A tip: you can represent a donut with a flat square in which each edge wraps around to the opposite edge.

solution posted