Knight Returns

[!IMPORTANT] Model the board as a graph: squares are vertices and legal knight moves are edges. Then a knight return in exactly $k$ moves is a closed walk of length $k$ from the starting vertex.

Important properties

• A knight move flips square colour, so every closed walk has even length.
• One short even cycle is enough to construct infinitely many larger even closed walks by repetition.
• The parity obstruction is global: it does not depend on the start square, only on the colour-switching rule.
• Construction questions and impossibility questions are two sides of the same graph model.

Typical proof pattern

1. Rule out odd $k$ immediately by colour parity.
2. Exhibit a concrete even closed walk.
3. Repeat or concatenate even closed walks to reach the required length.

For the large values such as $2025$ and $2026$, the size of the number is not the real issue. The issue is whether its parity matches the graph structure.