Best mathematical insight
On this site we often see answers in languages specifically designed for short code, or designed to be fast. Sometimes, a nice golfing trick or speed-up technique surprises us with its ingenuity, beyond the standard use of that language.
And occasionally an answer shows up that uses an unexpected approach to greatly simplify the problem, and makes us wonder how the author could ever think of that. This usually involves some far-from-obvious mathematical equivalence, or a particularly simple approach to the problem that was not evident at all (once revealed, other answers often follow the same approach).
This category is for the answer with the best mathematical insight or unexpected approach that led to greatly simplifying the problem, in any challenge type (code golf, fastest code, or others). The insight should have led to a significant improvement according to the challenge's metric (code length, run time, or whatever applicable).
Nominated by Bubbler
To quote xnor's comment, "My proof is an epic journey through plane geometry, Eisenstein integers, factorization over number fields, quadratic reciprocity, arithmetic progressions, and interchanging summations -- all for such a simple expression."
It was so epic that he didn't give the proof himself; instead he put a +500 bounty for the proof that his solution is correct. Lynn came up with a proof, and you can read Peter Taylor's more self-contained proof too. As the one who wrote this challenge, I'm still amazed when I read these proofs.
Incidentally, xnor's solution is the shortest among all solutions written in any practical programming language, and it even beats two Mathematica solutions.
Nominated by Bubbler
Unlike the solution on Triangular Lattice Points (which uses deep knowledge in number theory), this one is pure logic and careful analysis of the problem that led to very short and elegant solution. The astounding fact is that this is short and fast; by eliminating a branch of recursion, it achieves linear time complexity from an otherwise exponential algorithm.
Nominated by Kevin Cruijssen
The challenge states the following: "The sequence contains the decimal representation of the binary numbers of the form: 10101..., where the n-th term has n bits."
Reading this, the first approach that comes to mind is using the binary representations themselves. But no, @Neil's insight and approach was simply brilliant, without using any binary builtin at all. By using \$\frac{2}3\$ (having a binary representation of 0.101010101...
), multiplied by 2 to the power of the input-integer n
, his approach was not only mathematically genius in its simplicity, but even short enough to be used by multiple other answers (even golfing languages like 05AB1E) to save bytes.