Yesterday we did a fun project on a problem I learned from Michael Pershan
That project is here:
Sharing a problem from the Julia Robinson math festival with the boys
Last night I got an interesting comment on twitter in response to my Younger son suggesting that we write the numbers in a circle – a suggestion that we didn’t pursue:
So, today we revisited the problem and wrote the numbers in a circle:
Next I asked them to try to find another set of numbers that would lead us to be able to pair all of the numbers together with the sum of each pair being a square. The discussion here was fascinating and they eventually found
This problem definitely made for a fun weekend. Thanks to Michael Pershan for sharing the problem originally and to Rod Bogart for encouraging us to look at the problem again using my younger son’s idea.
Mike's Math Page 
You can also use pythagorean triples to construct pairings, building off the pattern from 16.
So, there are (at least two) recipes for building pairings:
1) given a pythagorean triple (a,b,c) with a< b, the numbers 1 to a^2-1 will pair to sum a^2, the numbers a^2 to b^2 will pair to sum c^2
2) iterative recipe: given pairings for 1 to n (with the largest sum equal to m^2), then we extend to (m+2)^2 – n – 1