Something Mildly Clever

I woke up in the middle of the night last night and couldn't manage to fall back asleep, so I started thinking about how I would prove the Four Color Theorem. (This is one of many little tricks I use to disctract myself from whatever is really bothering me and keeping me awake. It almost never works, but if I am going to be up I might as well think about something like this or Fermat's Last Theorem or non-orientable surface integrals or something like that).
(Yeah, I'm weird like this.)

I started by trying to divide up an arbitrary map using Poincare Maps. I quickly decided I didn't remember enough about Poincare Maps to have any chance with this. :(

Unfortunately, it didn't put me to sleep either.

So then I thought about Euler networks and tried to figure out if I could figure out the Euler characteristic of simple maps and expand outwards. I pondered this for about an hour, decided I couldn't get to fewer than 5 colors and finally fell asleep.

In looking up the wiki on poisson distributions below, I remembered my problem last night and try to see if there was any merit to what I was trying.

Is seems there was. :)
(although I was not as concise as the wiki version)

It's a little reassuring to know that the brain worms haven't gotten everything yet.