Square Sum Puzzles

A couple nights ago, my son was finishing up his nightly Beast Academy and enjoying the puzzles so much that he took his workbook with him at bedtime and continued to work for another hour!

He was in a section of the book on multiplication, and these particular problems asked students to arrange given numbers so that the sum of any two consecutive numbers is a perfect square. For example, if the given numbers were

then one solution would be

After a handful of warm-up problems like these, we came to the main attraction:

 

(Apparently, among the people who know puzzles like this one, this one is very well-known.) I’ll let you find the solution yourself.

This puzzle is perfect for several reasons. First, it works. Puzzles without solutions are, quite literally, a waste of time. Second, it’s unexpected. Perfect squares are special integers, so it feels like the numbers given in the blocks should have to be specially chosen somehow. How could it possibly be true that every number up to $16$ fits? Finally, (and you’ll only understand this point if you solve the puzzle yourself), it’s extremely doable. My 8-year-old came up with the strategy to solve it all by himself, and the answer just sort of rolled out.

I was so excited about this puzzle that I tried to see if I could come up with more examples, but I always got stuck. Sometimes, the puzzle wouldn’t have a solution. For example, there is no solution to the $1$-to-$8$ version of the puzzle.

Other times, there is a solution, but finding it is a bit messy. To see what I mean, try the $1$-to-$32$ version of the puzzle. The answer is nice, but it can feel a bit tedious to work out.

(Some people have explored other avenues of this puzzle. Apparently, the numbers from $1$ to $305$ can be rearranged so that each consecutive pair adds up to a perfect cube. There are also variants where the sums of consecutive numbers are primes instead.)

So, I hope you enjoy this puzzle as much as my son and I did. It doesn’t get any better.