I'd love some background on the problem - I'm a big fan of well-explained proofs like this, but I'm not a mathematician by trade and I like hearing a little bit about why anyone is thinking about this stuff in the first place.
Why should it be that a doodle is always two-colorable? It's a question a child could ask, but it leads to the whole area of graph theory. Personally, I find it a much better introduction to Graph Theory than the usual question about the Bridges of Königsberg:
https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
But there is a progression.
* A doodle is two-colorable
* A doodle is a planar, connected, Eulerian graph
* The dual of a planar, connected, Eulerian graph is bi-partite
* Bi-partite graphs are trivial to identify, and trivial to colour.
* What about tri-partite graphs?
* Identifying whether a given graph is tri-partite is NP-Complete.