# We Got A Problem #5: Tiled Pools

November 4, 2011 7 Comments

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!*

“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.

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…)

Ah. Zing!

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.

Al

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

Is it possible to use exactly the same number of white and black square blocks? Is it possible to do what with the same number of white and black squares? Please clarify.

The pool (picture an interior rectangle) is made of white square blocks, while the border (like a picture frame) is made of black square blocks. The goal is to find dimensions of the rectangle so that the number of white and black blocks are equal.

For example, picture an 8-by-10 pool with a single block of black as border. Adding the border creates a 10-by-12 rectangle.

Pool (white): 80 tiles

Border (black): 40 tiles

So, an 8-by-10 pool with a single block of black border is not a solution. Hope this helps!