This is a long one and took quite a bit of work in the end, but I hope you enjoy it. It's a story of problem-solving, centered on one quite fun puzzle, which is to find the average area for the shadow of a cube, averaged over all its possible orientations.

## Alice, Bob, and the average shadow of a cube

## Alice, Bob, and the average shadow of a cube

## Alice, Bob, and the average shadow of a cube

This is a long one and took quite a bit of work in the end, but I hope you enjoy it. It's a story of problem-solving, centered on one quite fun puzzle, which is to find the average area for the shadow of a cube, averaged over all its possible orientations.