Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

This is just one of twelve problems posed to 24 UML undergraduates as part of the 82nd annual William Lowell Putnam Math Competition sponsored by the Mathematical Association of America on December 4. Thousands of students throughout the United States and Canada simultaneously worked on these problems with no technology other than a pencil and paper. There were two 3-hour sessions with six problems in each session.

The Putnam returned to campuses after an unofficial virtual competition in early 2021. Professor Kenneth Levasseur supervised the competition at UML.

By the way, the solution to the problem above is $2-\frac{6}{\pi} $. You have to show your work! All problems and solutions are available at the Putnam Archive.