I recently moved to a new house, and this house has more light switches than I'm used to.
There a N lights in the house and M switches where N < M.
A light may be controlled by 1 or more light switches.
If a light is controlled by more than one switch, then we'll say the set of switches which control that light are shared.
The reason for the sharing of switches is so that you can walk from one from to the next without needing to turn off the light while you are still in the room. But I keep forgetting which switches are shared, so I proposed to label them such that every switch that shares a light will have the same label.
Of course I could assign every light in the house a unique label, but I would like to use the fewest unique labels necessary.
Given a set of rooms, switches and their relationships, what is the smallest number of unique labels I will need such that no switch in a room shares a label unless they also share a light?
I should clarify that I actually did move recently and ran into problems with light switches which made me start thinking of this more general problem and I'm trying to find out how I would go about formalizing the problem and then finding a general solution for any house.
In actual fact you could use just one colour, and shape the stickers with a pair of scissors ( square, triangle circle, etc.) according to which light the switch belongs to. But this may not be in the spirit of the problem.
If you are asking this for a purely practical reason, then personally ,I would have all the switches that control the light in one room the same colour. If there is more than one light in a room I would still have the same colour for the switches but different shaped stickers for the different lights. And I would pick a colour according to a significantly coloured item in the room that is obvious, thereby making it easy to remember.