Another nice problem about dice that has a slick generating function argument: it is not possible to place weights on the sides of two dice (numbered 1--6) in such a way that every possible sum from 2 to 12 occurs with probability 1/11.

So many good options here! But I'll just post one:

There's a very beautiful and short generating function proof of the following result: There is no nontrivial partition of the nonnegative integers into arithmetic progressions with distinct moduli.

Our reward for accepting them is that they all come together so spectacularly into one formula, thus showing us that we were right to accept them.

Many people find Euler's identity e^{iπ}=-1 or e^{iπ}+1=0 to be the most beautiful formula in mathematics. To me, this identity is a triumph of bravery: every ingredient in this formula was a controversial idea at one point: zero, negative numbers, irrational numbers, imaginary numbers.

But there is a positive takeaway, which is that it encourages students to accept that their world can grow. If you don't have an object with properties that you want, then you can just make one. Of course it might not have good properties elsewhere, but the idea may be worth exploring.

I believe that this is a harder pill for students to swallow than limits are, in that it requires a greater strain on their sense of reality.

Also, every book title has to start with "An Introduction to" or "A First Course in" or "Elementary."

This is one reason you should never use an indefinite integral. Instead, always use a definite integral, possibly with variable bounds.

Come back, both of you!