Sometimes, after solving a problem, it seems to me there should be a symmetry argument that I missed.
You have a cup of coffee and a cup of cream. Put a teaspoon of the cream in the coffee and stir. Take a teaspoon of this mixture and put it in the cream. Is there now more cream in the coffee or coffee in the cream?
Neither. It turns out that they are equal. Should this have been obvious? Why? Is there a symmetry argument?
Ann says: Man! that took me a while to figure out. I forgot that the volume of the coffee before transferring back to the milk is (T + 1)/T. So in the end, each cup has 1/(T + 1) of the other?
I think that's right. If the spoon holds T cups, then in the end the coffee cup contains 1/(T+1) cups of coffee and T/(T+1) cups of cream. Vice versa for the cup of cream. You can stick in T=0 and T=1 and see that the formulas make sense. After doing the math though, it seems that there should be some way to see intuitively why the equality is there.