I’ve Got a Problem!

Many, many years ago, I saw this problem somewhere:

Arrange the whole numbers from 1 to 18 into nine pairs, so that the sum of the numbers in each pair is a perfect square.

I liked the problem, and included it in a book I co-authored (Algebra: Themes, Tools, Concepts, following lesson 5.5). In the Teacher’s Edition, I suggested that this could be a “problem of the week,” and that a “challenging generalization” was to figure out whether this could be done with numbers other than 18. I then promptly forgot about the problem.

More than 20 years later, last August, I was asked to participate in the training of a group of graduate students who were going to lead Math Circles in San Francisco. My topic was “creating a problem-solving culture”, and it seemed like using a sample problem would be helpful, in order to root my pedagogical advice in something specific. (Presentation slides)

After the participants in the workshop had worked on the problem for a bit, I interrupted them, and asked why this was a decent problem. Some answers:

• not a lot of prerequisites (knowing small perfect squares, being able to add)
• the answer is not obvious
• partial solutions are possible (e.g. one can find eight pairs that satisfy the constraint) 
• insight increases along the way
• the problem can be generalized

The third point suggests a rewording of the problem:

Arrange the whole numbers from 1 to 18 into nine pairs, so that the sum of the numbers in as many pairs as possible is a perfect square.

This is a much better problem, because students can immediately find solutions, and a healthy competition develops about finding solutions with more and more pairs.

When some students get nine pairs, they can work on the generalization:

Find some numbers other than 18 
for which this is possible.

When I got home, I got interested in the generalization, and over the next few days I worked out solutions for half the even numbers from 2 to 28. The other half was not possible. As the numbers got larger, the calculations got more tedious, and false starts took more and more time, so I decided this was a good opportunity to write a program in Python, a computer language I have been vaguely trying to learn since last summer.

(Continued in my next post: Getting Help)

–Henri