You should check out Alexandre Muñiz' Puzzle Zapper Blog. Muñiz invents all sorts of attractive and clever puzzles. There is a lot to like about these puzzles even if you don't particularly care for puzzles. They always have some elegant mathematical symmetry and an interesting concept. Puzzle Zapper also has all kinds of things things related to tilings, geometry and combinatorics.
Today's post is about which pentominoes are the most convex. Obviously, the I pentomino is and the others aren't, but how much aren't they? Well, it depends on how you measure. By reasonable method A, it's the U pentomino; by method B it's the X, and by method C it's a tie between F, T, V, X, and Z.
It's easy to show, and has been known for thousands of years, that this is the only way (not counting rotations and reflections, of course) to arrange the numbers 1–9 in the cells of a 3×3 array so that the numbers in each row, column, and diagonal add up to the same sum, which must be 15. Muñiz arrived at the Gathering and anounced that he had found not one but two new ways, and the story just keeps getting better from there.
My only regret is that his posts are so infrequent. But every time I see there is a new one I smile and exclaim “Ooh, a new Puzzle Zapper!” and rush off to read it.