From Michael Branicky comes a puzzle that was inspired by an email thread from James Propp:
Suppose you roll a fair six-sided die on a ridged gnocchi board, such that two adjacent faces come up every time.
On average, what is the sum of the numbers shown on those two faces?
There are 12 edges on a die.
There are three pairs of opposite faces on a die: 1 & 6, 2 & 5, and 3 & 4. All opposite face pairs sum to 7.
Landing on an edge corresponds to one side from each of two out of the three pairs.
Since each is equally likely, the average sum is 7.