A few years ago, I directed a high school summer math program. Half the day was devoted to exploring the delights of modular arithmetic—we ended the summer with a cake decorated with Fermat’s Little theorem!—and half to learning to program in Python, with number theory questions as motivation. One Friday afternoon, we included these questions in the programming part of the day.

Goldbach conjecture: Any even number larger than 2 is the sum of two prime numbers.

- Is there a counterexample to this conjecture for an even number less than 10,000
- Prove this conjecture.

Collatz conjecture: Choose some number a_{0}.

Define a_{n}by a_{n}=3a_{n-1}+1 if a_{n-1}is odd or a_{n-1}/2 if a_{n-1}is even.

Then a_{n}will be 1 for some n.

- Is there a counterexample to this conjecture for a
_{0}<10,000?- Prove this conjecture.

Perhaps it was a tiny bit evil to give these longstanding open problems to high school students without warning them, but it was a lot of fun to watch them come up with programs to search for counterexamples and brainstorm about ways of approaching the proofs. (And yes, we did eventually tell them the questions were still open. We didn’t want to ruin their weekends completely!)

Math teachers Annie Perkins and Sam Shah have written about the benefits of exposing kids to advanced math concepts early rather than waiting until they’ve mastered all the easy stuff. If you too would like to torture your students kindle your students’ curiosity and challenge their intuition, with unsolved math problems, there are lots of places to go for inspiration.

The MathPickle site (tagline: “Put your students in a pickle!”) has puzzles organized by grade level, board game suggestions, and a blog. I’ve seen this site mentioned in a few places, including a discussion on Dan Meyer’s blog.

Lior Patcher has a list of suggestions for how to use unsolved problems in K-12 classrooms at his computational biology blog Bits of DNA. I was especially excited to see a question involving Namibia’s mysterious “fairy circles,” circular patches of bare ground surrounded by vegetation. It’s nice to see some modeling and applied math get some love there. Why should number theory have all the fun?

Mike Lawler often discusses advanced and unsolved problems with his kids, and the Collatz conjecture has made several appearances on his blog. In his most recent post on the topic, his kids make music with John Conway’s “amusical” variation of the problem. (As a violist, I’m delighted that one of them does so in alto clef!)

Ben Braun writes about using unsolved problems in his college math classes at the AMS math education blog On Teaching and Learning Mathematics. He highlights some of the benefits for his students, including mindset shifts away from answer-getting and toward seeing failure as part of mathematical productivity.

It goes without saying that your students probably won’t solve the Goldbach conjecture or get a definitive answer about fairy circles in one or two class periods, but you never know. Some open problems might end up being easy to solve, or at least easier than we might think. The Gödel’s Lost Letter and P=NP blog has a fun post about open problems with short solutions.

Several years ago Tim Gowers did a very long post relating to math pedagogy, with LOTS of ideas, that may interest your readers:

https://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

I wrote an article about 10 years ago describing my undergrad research seminar at Illinois, which helps teach math majors how to come up with their own questions and attempt to solve them:

“One Introduction to Mathematical Research”, Math Horizons Vol. 14, No. 2 (November 2006), pp. 16-17, 38.

A copy can be found at http://www.jstor.org/stable/25678652 or https://faculty.math.illinois.edu/~reznick/mhoriz.pdf

Particularly valuable resource at MathPickle is the list by grade of unsolved problems from a conference Gordon organized. http://mathpickle.com/wp-content/uploads/2016/01/Unsolved-K-12-winners.pdf

He mentions another collection by Lior Prachter: https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/#comment-4987

Whoops! Don’t know why I missed your mention of Lior on first read.