# Puzzle: Clarence the Clever Contractor

No, no relation to Joe the Plumber.

Clarence the Clever Contractor cleared a rectangular plot of land and covered it with gravel. Then he purchased 9 square wooden sections of side-length 2, 5, 7, 9, 16, 25, 28, 33, and 36. By placing the squares on the gravel with no two overlapping, Clarence built a patio which exactly covered the graveled surface. Find the perimeter of Clarence’s new patio.

If you find this easy to solve, why not leave hints, or comments instead of the answer? And if you don’t find it so easy, please, ask questions, share the directions you are exploring.

I took this from my graduate class, years ago, and then dropped. A colleague brought it back to me, and I started using it again. I don’t know where the professor borrowed it from, but I am certain it was borrowed.

You say 9 square section, but only list 8 values… is there one missing? Or does this mean 9 sections EACH of 8 different sizes for a total of 72 sections?

The 9th value was 9 – I added it back in. Thanks for the catch.

I like the way the numbers work better with that 9 back in there. :)

My hint is to find the total area of the 9 patio squares, and then find its prime factorization.

Since I’m on a prime factorization kick with my middle schoolers these days, perhaps I’ll bring this in as a bonus problem.

Here’s a question: I knew what the answer had to be before I knew if it would really work or not. So I then drew out the picture to assure myself that all the pieces would indeed fit nicely.

We often assume that there has to be an answer because of the way the puzzle was posed. In this case there was only one plausible answer, so I could have probably just assumed it was “the” answer without going any further (and certainly would have done so in a time-limited situation).

I can see that it would sometimes be possible to prove that no answer was possible, but is it generally possible to prove that a plausible answer is possible without drawing the picture?

In fact, the first time I ‘solved’ this I was left fairly upset… I knew what was good about my answer, but I really didn’t know if that solution existed.

I’ve had kids ‘build’ the patio – seen it with my own eyes – so yes, the solution is good. But you are right. How are we supposed to know?

I started at the center and built outward. Mentally, I was thinking sunflowers. Fun problem.

I agree – nice problem. I solved it Math Mom’s way. Didn’t draw the picture – I saw there were two different ways, and one was implausible.

I’m glad you liked it.

Does anyone recognize this one? Know where it came from?

I have a bunch of them that I think I got from Erich Friedman place

http://www.mathpuzzle.com

Find the total area and prime factor it? Wow. I saw it differently and so clearly – I was working out from the middle – that it took me a couple of thoughts to even figure out what you were trying to do with that! Talk about mental tunnel vision.

I saw this as an additive combinations problem and I had most of the combinations in a few seconds …

2+ 5 = 7, so the 7 sidles up to the 2-5 stacked

7 + 2 = 9, so the 9 rides on top of the 2-5 / 7 leaving 3

which had to be the difference between the 36 and 33

7 + 9 =16, so the 16 sidles up next to the 7-9 stack

and so on.

I have another from an Informal Geometry book, which drew out the squares, labeled the 7 and 9 and left finding the rest of the sides for the students.

7, 9, 16, 19, 26, 28, 33, 44, 45, 60

And the power of this problem is what happens in the classroom: some groups of kids attack the area, others attack the model, and they align the two correct, but thoroughly distinct solutions.

In fact, with factoring, a second solution presents itself, but the model blocks it. I make sure that students have a chance to inspect and reject the erroneous solution.

Nice one.

TRM thanks for sharing that link.