To me, going to bed often feels more like a tiresome deprivation from life than a welcome rest, or a painless detour through oblivion to morning. When I lack patience for it, I like to think about math puzzles. Other purposeful lines of thought keep me awake or lose me, but math leads me happily into a world of abstraction, from which the trip to dreamland comes naturally.
(It doesn’t always work. Once I was still awake after seemingly solving two Putnam problems, which is about as well as I did in the actual Putnam contest.)
A good puzzle for this purpose should be easy to play with in one’s head. For me, that means it should be amenable to simple visualization, and shouldn’t have the kind of description you have to look at multiple times. A handful of blobs is a great subject matter; an infinite arrangement of algebra is not.
Recently I’ve been going to sleep thinking about the following puzzle. I got several nights of agreeable sleep out of it, but now I think I have a good solution, which I’ll probably post in future.
Suppose that you have 1 kg of red clay that is 100 degrees and 1 kg of blue clay that is 0 degrees. You can divide and recombine clay freely. If two pieces of clay come into contact, temperature immediately equilibrates—if you put the 1kg of red clay next to 0.5 kg of blue clay, all the clay will immediately become 66 degrees. Other than that the temperature of the clay doesn’t change (i.e. no exchange with air or your hands, no radiation, etc.). Your goal is to end up with all of the blue clay in a single clump that is as hot as possible. How hot can you make it? (Equivalently: how cold can you make the red clay?)
HT Chelsea Voss via Paul Christiano
Clue: it’s more than 50 degrees.