Puzzle time - truly even split

[jon-nyc]

Can you partition the integers from 1 to 16 into two sets of equal sizes so that each set has the same sum, the same sum of squares, and the same sum of cubes?

Trying to figure out if there’s a clever way to approach this without grinding it out or using a computer.

[Horace]

If it’s not 16 in partition 1, 15 and 14 in partition 2, 13 in partition 1, then repeat similar sequence of four three more times, then I doubt it is possible. I would have to do the math on that potential solution.

[Klaus]

I think it will be helpful to consider the binary representation of the numbers.

[jon-nyc]

I was able to get the answer from this morning’s hint:

HINT: It's easy to split the numbers 1 through 4 so that each part has the same sum.  Can you extend that to splitting 1 through 8, so that each part has both the same sum and the same sum of squares?

[jon-nyc]

The answer is not quite Horace’s suggestion but it’s close.

[jon-nyc]

I have a split for 1-32 that has equal sums, sum of squares, sum of cubes, and sum of 4th powers.

It probably scales indefinitely.