One of my favourite puzzles of all time comes from Paul Salomon. Paul is one quarter of the Math Munch team, and also makes the most beautiful mathematical art.
On Paul’s site he calls it ‘‘The Lost in Recursion’ Recursion‘ puzzle. I’ve retyped his chalkboard photo below:
In 2012 I mentored a mathematically keen Year 11 student. She and I worked on this puzzle for weeks before we figured it out together. Biting my tongue was so hard, but the payoff was worth it. So, I’m not going to tell you the resolution, but I’m happy to discuss (email, the comments, Twitter, in person).
When I started my ‘Developing Mathematical Thinking’ course for pre-service maths teachers, I included this in the list of possible project topics. Most students can’t work out what’s going on at the first glance so dismiss it as too hard. Last year two adventurous students (A and T), perhaps buoyed by my enthusiasm for the puzzle, decided to tackle it.
Except that I unknowingly reproduced it incorrectly in the project handout. A and T persevered, looking for patterns. I kept saying ‘keep going’, ‘what do you notice’, ‘you’ll make progress’ and talked through their ideas with them. They kept hitting dead-ends. Eventually, I sat down and took a good look.
The ease with which they took the news was both a relief and a surprise. (Perhaps they swore and cursed out of earshot.) I like to think that it was because I valued the process and not the final product. The aim of the project is to have an authentic research experience — to play, explore, discover, conjecture, reason — just like professional mathematicians. And that’s what happened, in both versions of the puzzle1.
 The title of this post is inspired by the title of their final project report ‘Unravelling recursive calculations’. Their final presentation made me so proud; they worked so hard to successfully explain a complicated puzzle to those peers who had dismissed it as too difficult.