For the Dim Bulbs
You rustle up a few pals and get to work, and a few hundred dollars later the place is looking like less of a dive, but in a moment of belated clarity you realise that only one thing needed to be changed to cement its 'cool' status:
just install 'The Clapper' in every room in the house.
For those too young to remember the Clapper heyday, this device controls the on/off switch to a lighting source, and toggles its position when you clap in the audible range--which, for added value and amusement, tends to extend at least into adjacent rooms. For our purposes, let's say the device's range is exactly its host room and those directly adjacent. So, say you've got one light source per room, each with a Clapper, in a house which, like any good house, is actually just an m-node undirected graph.
Now, for your grand house-rewarming party, you invite all your friends over, wait until their thinking skills are nice and clouded, and issue a challenge: clap the house into a fully-lit state.
In fact, no matter what your house's structure, this challenge can always be met, as long as lights are initially off! Can you prove it?
I came up with this result a few years ago after playing a computer solitaire game, which presented this problem for a 5-by-5 square grid (don't recall if there were diagonal connections... anyone have a weblink?). Thought I was pretty clever, but then I saw it given as a problem in an old math journal, I think an AMS one (I'll post a ref if I ever find it). No fancy tools needed to solve this fairly simple puzzle, although of course some linear algebra would help.
Extra Extra Credit: Now some of the lights are off/bright, while some are off/dim/bright. Now it's no longer always possible to get all lights bright simultaneously. In fact, I'm guessing it's NP-complete to maximize the number of bright lights, but haven't proved this. What's the story?
Oh, and the wasabi-green tip came to my attention (briefly!)in a recent New Yorker.