Problem
COM-B1-M07-P016 Closed Knight Tour
#16
★★★☆☆ Level 3 of 5
Prove that on a \(5\times5\) board there is no closed knight tour visiting every cell exactly once.
A closed knight tour would be an odd cycle in a bipartite graph.
Color the board as a chessboard. Every knight move changes color, so the knight graph is bipartite: all edges go between the two colors. A closed tour through \(25\) cells would be a cycle of length \(25\). But every cycle in a bipartite graph has even length, because colors alternate. Contradiction.
Bridge to the graph module.