Question 189
Let $n \geqslant 2$. A circuit is drawn on an $n \times n$ chessboard that passes exactly once through each square and returns to its starting point (two consecutive squares on the circuit must share a side).
Show that there exist two neighboring squares on the chessboard such that, if we "cut" the circuit at these two squares, the two paths obtained have length at least a quarter of the length of the original circuit.
Show that there exist two neighboring squares on the chessboard such that, if we "cut" the circuit at these two squares, the two paths obtained have length at least a quarter of the length of the original circuit.