It seems that the readers of this blog like puzzles. There is circumstantial evidence for that, so being a theorist, I will declare that to be true.
I thought I would start the year with one such puzzle. Here is a typical interesting probability problem that is posed for amusement. Assume that you have a stage (I’ll use cartesian coordinates and put it at (0,0)) and that people can be seated anywhere on a rectangular grid (the locations with integer entries). It could also be a honeycomb grid. In fact, any lattice will do. You can also assume that the arena where this is taking place has radius N where N is large. An important question that you might ask, is if you can see the stage when you are seated at some location. For simplicity, let us assume that you can see the stage if the straight line from your location to the origin hits no other head (another integer lattice point). If the show is packed, meaning all locations are full, what is the probability that you will have an unobstructed view?
At first sight, it might seem like that probability is very low when N is large,
but the answer is actually finite in the large N limit.
I will give you a non-standard physics-like `proof’ of the correct answer in a subsequent post that will generalize to higher dimensions immediately. I happened on this proof one morning in late December while I was in the shower and this problem popped in my head (I’m not sure why). Since I couldn’t remember the answer precisely, I ended up proving it again. This is also a famous problem in number theory if you look at it the right way. In the meantime, I will leave the puzzle on to make you think about it for a while (in case you don’t already know the answer).