Out of the problems on the app, it has been proven you can get up to 8 for original (check all cases), infinite for odd (see below), 16 for even (as amount for original doubled), 24 for multiples of 3 (as amount for original tripled), and infinite for non-multiples of 5 (see below), cubes (see below), prime (see below), and fibonacci (see below).

Proofs:

For odds:

If you put all the odds on one side, then it will work because odd+odd=even.

For non-multiples of 5:

Put all the numbers ending with 1,4,6,9 on one side and all the numbers ending with 2,3,7,8 on the other side. For the last digit sums, 1+4=5, 1+6=7, 1+9=0, 4+6=0, 4+9=3, 6+9=5 so the first side is safe. 2+3=5, 2+7=9, 2+8=0, 3+7=0, 3+8=1, 7+8=5, so the second side is also safe.

For cubes and nth powers, see this:

For primes:

Put them on one side except if the number is the second member of a twin prime (for 3,5,7, put 3 and 7 on one side and 5 on the other). It will work because of the odd theorem and because the gap between the numbers on one side must be greater than 2.

For fibonacci:

Put them any way without having 3 in a row on one side.

But squares remain unsolved. Please share if you have any results or questions!

- aj

Thank you AJ, lovely work!

My break-out room focused on squares. We found that when a square was a member of a Pythagorean triple (25, for example is the sum of 3^2 + 4^2) then it would be forced on the opposite leaf. When we found a square that was a member of

twoPythagorean triples, that was when things got tricky (625, for example is the sum of 7^2 + 24^2and15^2 + 20^2).We weren't able to come up with a cohesive proof, but it felt like, with some planning we could guarantee that the rare numbers that were members of

two Pythagorean Tripleswould have a leaf to land on if we could split up the smaller squares.For the 625 example, as long as 7^2 and 24^2 were on different leaves, then 625 would have a home.

I wonder if you could take this starting point and explore it some more!

We found a resource:

http://nonagon.org/ExLibris/fermat-sum-two-squares-calculatorto quickly check whether a number was a sum of squares and what those smaller squares are.