I wish I was closer. I like to solve mysteries.
Have you opened up every wall and ceiling box in the room and surrounding rooms to see if you can tell where they go?
Be careful. 110 can and will kill. Probably as quick as a 10mm!
Since they only have one cable, they obviously go either...
I would recommend a Fluke LVD1 voltage detector. A non contact voltage detector is a very handy tool. You just need to know what it is telling you and when you must verify with other means.