Pesachim 112b relates a story about demons. The queen of demons (with a large army) used to be permitted to wreak havoc at will (which is a separate question), until one day when she encountered R. Chanina b. Dosa. She told him that they spoke of him in heaven and told her of his learning, so she was not allowed to take action against him. He availed himself of this opportunity to forbid her to travel through settled areas, but when she pled for mercy he relented and said she could come out on two nights, Tuesday night and Shabbat (Friday night).
Why those particular nights? Shabbat in particular seems like a peculiar choice to me; it seems like a time that we would want to minimize, not maximize risk, since it is a social evening. And Tuesday night just seems random. Furthermore, it seems rather odd that he would give her two nights, and not just one, or even that he would relent at all. Is there something we're supposed to learn about those nights from this episode?