It’s Not What’s on the Outside…
Through the Academic and Creative Endeavors (ACE) program at their school, my sons participate in the Math Olympiad for Elementary and Middle School (MOEMS). While passing the door to the ACE room yesterday, I noticed a sign with the names of those who scored a perfect 5 out of 5 on the most recent contest — and my sons’ names were conspicuously absent. Last night at dinner, I asked Alex and Eli what happened, and they told me about the problem that they both missed. (What? Like we’re the only family in America that discusses math problems at the dinner table.) Here’s how they explained it to me:
Nine 1-cm by 1-cm squares are arranged to form a 3 × 3 square, as shown below. The 3 × 3 square is divided into two pieces by cutting along gridlines only. What is the greatest total (combined) perimeter for the two shapes?
The answer to this problem appears below. Pause here if you’d like to solve it before reaching the spoiler.
[Ed. Note: I didn’t see the actual exam, so the presentation of the problem above is based entirely on my sons’ description. Apologies to MOEMS for any substantive differences.]
This problem epitomizes what I love about math competitions.
- The answer to the problem is not obvious. This is the case with many competition problems, unlike the majority of problems that appear in a traditional textbook.
- The solution does not rely on rote mechanics. Again, this differentiates it from a standard textbook problem or — shudder! — from the problems that often populate the databases of many skill-based online programs.
- Students have to get messy. That is, they’ll need to try something, see what happens, then decide if they can improve the result.
- Students have to convince themselves when to stop. Or more precisely, they’ll need to convince themselves that they’ve found the correct answer. For instance, let’s say a student divides the square into a 3 × 2 and a 3 × 1 rectangle. The combined perimeter is 18 units. Is that good enough, or can you do better? This is different from, say, a typical algebra problem, for which students are taught how to check their answer.
The problem also epitomizes what many people hate about math competitions.
- There’s a time limit. Students have 26 minutes to solve 5 problems. Which means that if students spend more than 5 minutes on this one, they may not have time to finish the other four. (There was a student of John Benson who, when asked about his goal for an upcoming math competition, replied, “I hope to solve half the problems during the competition and all of them by the end of the week.” That’s the way mathematicians work.)
- It’s naked math. Sorry, nothing real-world about this one. (But maybe that’s okay, because real may not be better.)
- The problem is presented as a neat little bundle. This is rarely how mathematics actually works. True problems often don’t present themselves all at once; it’s through investigation and research that the constraints become known and the nuances are revealed.
All that said, I believe that the pros far outweigh the cons. Benjamin Franklin Finkel said, “Many dormant minds have been aroused into activity through the mastery of a single problem.” I don’t remember the last time a mind was aroused by the solution to x + 7 = -3, but I’ve witnessed awakenings when students solve problems like the one above.
And here’s the tragedy in all of this: Many teachers believe that only kids who participate in math competitions can handle — or appreciate — math competition questions. No! Quite the opposite, in fact. Students who have tuned out have done so because they’ve never been challenged and, worse, have never felt the thrill of solving a problem on their own.
What I really love about the problem, though, is it made me think about other questions that could be asked:
- How many ways are there to divide a 3 × 3 square into two pieces that will yield the maximum total perimeter?
- What is the maximum total perimeter if a 3 × 3 square is divided into three pieces?
- What is the maximum total perimeter if a 4 × 4 square is divided into two pieces? …a 5 × 5 square? …a 6 × 6 square?
- The answer to the problem about the 4 × 4 square appears below. Pause here if you’d like to solve it before reaching the spoiler.
- (wait for it) What is the maximum total perimeter if an n × n square is divided into two pieces?
It was that last question that really got the blood pumping.
Here’s a solution for how to divide a 3 × 3 square to yield the greatest total perimeter:
And here’s a solution for how to divide a 4 × 4 square to yield the greatest total perimeter:
What’s the solution for an n × n square? That’s left as an exercise for the reader.