Alternations

These problems are based on simple observations on adding even or odd numbers. Let us start with a simple problem.

Problem1. Can a knight start in the (a1) square of a chess board and go to (h8) ( the opposite square on the diagonal ) and visit each of the remaining squares only ones?
Answer. No since we need an even number of moves to stay in the same color. We need to make 63 moves for that and the knight changes the color in which of the moves.

Methodological Remark. Many problems deal with proofs that certain situations are impossible. This poses some difficulty for mathematically naive students. Their first reaction is either frustration that they cannot find the correct situation (fulfilling the impossible conditions) or a declaration that the situation is impossible., without a clear conception of what it might take to prove this.

Problem2. A closed path is made up of 11 line segments. Can one line, not containing a vertex of the path, intersect each of the segments?