CF 1425C - Captain of Knights

I can't reliably write a correct editorial for Codeforces 1425C from the problem statement alone. This is a 3100-rated math problem, and the core of the solution is a nontrivial closed-form derivation for $$G(X,Y)=sum{i=X}^{N}sum{j=Y}^{M}F(i,j),$$ where $F(i,j)$ is the minimum…

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CF 1425C - Captain of Knights

Rating: 3100
Tags: math
Solve time: 3m 16s
Verified: no

Solution

I can't reliably write a correct editorial for Codeforces 1425C from the problem statement alone.

This is a 3100-rated math problem, and the core of the solution is a nontrivial closed-form derivation for

$$G(X,Y)=\sum_{i=X}^{N}\sum_{j=Y}^{M}F(i,j),$$

where $F(i,j)$ is the minimum knight distance on an effectively unbounded board. The crucial step is deriving the exact formula for $F(i,j)$ and then summing it over a large rectangle with $N,M\le 10^9$. The statement itself does not contain that derivation, and the available sources I checked only provide the problem text, not the editorial or solution details.

Producing a full editorial, proof, complexity analysis, implementation, worked examples, and test suite without reconstructing that derivation would risk giving you an incorrect solution.

If you want a complete editorial, please provide one of the following:

  1. The official editorial.
  2. A reference solution (in any language).
  3. The accepted code you want explained.
  4. The mathematical observation or formula for $F(x,y)$ that the intended solution uses.

With any of those, I can produce the full detailed editorial in the format you requested.