We Got A Problem #5: Tiled Pools

Lots of math books have problems about tiled pools, and they always seem to have a tiled border.  Area and perimeter, and all that.  (Never mind that the tiled border isn’t the perimeter, because of the corners…)

Picture a pool (okay, a rectangle) made of white square blocks, surrounded by a border of black square blocks.  Is it possible to use exactly the same number of white and black square blocks?  If so, find all the ways.  If not, prove it can’t be done.

Bonus points for creating an interesting 30-second video to motivate this exact question and no others!

More bonus points: think of some other questions that extend this problem, and let everyone know.  There are lots of interesting ones!

(Thanks to Matt Chedister and the PROMYS for Teachers crew for bringing up this interesting problem.)

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!