Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
This puzzle comes from one of my favourite resources, nrich.maths.org. That site is a treasure-trove of rich low-threshold high-ceiling tasks. I’m not going to explicitly tell you how to solve it here — you can read about it at nrich — but the answer will be revealed, so stop now if you want to try the puzzle first.
I gave this puzzle to my pre-service teachers. I’m always fascinated by the variety of ways students tackle puzzles. For this one, most students decide to use 17 pieces of paper (or 17 sticky notes, because who doesn’t love sticky notes) to make it easier to shuffle the numbers around.
Some just jump straight in and experiment. If they find a solution with this trial-and-error approach, I ask them whether it is the only solution — this encourages them to explore a more systematic approach. Others start by making a list of square numbers. This leads to quick discussion about the smallest and largest squares it is possible to make, and how some of them will need to appear more than once.
Eventually students decide that they need to find all pairs of numbers from 1 to 17 that sum to square numbers, and organise them in some way: tables, lists, matrices and diagrams. Some organise by writing them directly on their sticky notes. For example, the sticky note for 10 lists all the sums involving 10.
Some students are (over)-eager to share what they’ve done. Some are working more contemplatively. I wander over to a student — let’s call him Jay — who slowly but proudly reveals to me how he approached the problem. Jay doesn’t usually volunteer much, and I’ve been slowly coaxing more out of him. As I move away, I hear him quietly say ‘I’m not sure if it is important, but I noticed …‘.
I wheel around, eager to hear what he wants to share. What I heard almost knocked me over (well, not literally). And then I wrote it on the board.
Jay noticed patterns in the squares: 25,16,9,16,25. And the pattern repeats. Twice. I don’t understand (yet) why the pattern repeats, but it’s gorgeous.
Anything that a student wants to share is important to me — and often they notice the most interesting things that I hadn’t even seen.
Update (25 August 2016)
This week my 2016 class of pre-service teachers tackled this task. There were a few things that they noticed and wondered which I want to add to this post.
Along with the pattern in the squares described above (25,16,9,16,25 repeating), some students noticed that (mostly) the numbers from the left and right ends differ by 1. For example, at the ends, 17 and 16 differ by 1. Then 8 and 9 differ by 1. Skip 7. Then 1 and 2 differ by 1. Then 15 and 14 differ by 1. All the way into the centre. How cool is that?
Students wondered why they needed to use the numbers from 1 to 17. Why not some other highest number? One student conjectured that perhaps it works for 1 to (1+square number). Another student wondered if they could change the operation from addition to subtraction (or something else), do something similar. Or could they change the type of number that adjacent pairs should sum to? For example, I’ve just remembered this tweet from @MathsMastery. Lots of cool ideas to explore at another time!
— Mathematics Mastery (@MathsMastery) July 24, 2016