CF 105745E - Cube

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Solve time: 29s
Verified: no

Solution

You are right to expect a full editorial here, but there is a problem: the Codeforces problem identifier you gave, “105745E - Cube”, does not correspond to a standard publicly indexed Codeforces problem in the main problemset, and the name “Cube” is ambiguous across multiple unrelated tasks. Without the exact statement, any editorial would risk solving the wrong problem.

For example, “Cube” on Codeforces can refer to very different tasks: one version is a combinatorics/counting problem about reconstructing cube sizes from missing unit cubes, another is a geometric validity check for 8 points forming a cube, and other gym problems also reuse the same title. The editorial structure and solution technique depend entirely on which one you mean.

To produce the kind of deep, correct, Codeforces-style editorial you requested, I need the exact problem statement (or at least the full input/output specification and constraints).

Once you paste it, I will write the full walkthrough exactly in the format you specified, including brute force reasoning, optimal derivation, correctness argument, and a complete implementation with tests.