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!

About Bowen Kerins
Bowen is one of the lead authors of CME Project, a high school mathematics curriculum focused on mathematical habits of mind. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won $1,000 for knowing the number of degrees in a right angle.

5 Responses to We Got A Problem #5: Tiled Pools

  1. Dan Meyer says:

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

    I’ve read this line like five times and I can’t make sense of it. How do you provoke one question and no others? The only way to ensure a single question gets answered is to ask it, as far as I know.

    • Bowen Kerins says:

      I was just kidding with that line! There is absolutely no way to motivate one question and no others. Heck, even asking the specific question leads to more questions, like the 3-D version, or looking for pools where there would be exactly twice as many interior tiles as exterior.

      (The line was intended as a nod to the cool videos on Dan’s blog and the often fervent discussion that follows them…)

  2. Al Cuoco says:

    I was at the seminar when the teachers raised this. Almost immediately, Paul Fournier of Framingham (MA) High asked the same question about rectangular boxes made out of little cubes.


  3. Matt E says:

    I believe I have the answer, and am now thinking about “L-shaped” pools. It’s already a lot more complicated…

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