Knight's tour

A knight's tour asks a simple question with a stubborn answer: can a knight visit every square on a board exactly once, moving only as a knight moves? On a small board the question is approachable by hand. On larger boards it becomes a serious search problem. Closed tours, which return to the start, add another layer.

We treat this as mathematics, not as chess coaching. The knight's L-shaped move defines a graph: squares are vertices, legal jumps are edges. A tour is a path that visits every vertex once. Students do not need that vocabulary on day one, but the idea is already there whenever they ask which squares are still reachable.

Problems.cc has an interactive knight tour for this topic. Being able to place moves, undo, and see the remaining empty squares turns an abstract path problem into something students can feel. That interactivity matters because tours fail late: a board can look fine until the knight has nowhere legal left to go.

The knight's tour trains planning under branching choices. At every step there are several legal moves, and most of them are locally fine. The difficulty is global. Students learn that a choice which looks free early can strand them later. That is one of the cleanest introductions to path thinking we have for this age.

It also builds useful habits around representation. A child who only stares at a chessboard often loses track of visited squares. A child who marks a route, counts remaining squares, or notices corner-and-edge patterns begins to organise the search. Organising a search is a mathematical skill long before formal graph theory appears.

Parents sometimes worry that chess topics favour children who already play. In our classes that is rarely the issue. We teach the move once, then focus on route, stuck positions, and recovery. The prize is not opening knowledge. It is the ability to think ahead, revise a plan, and explain why a path died.

We start on boards small enough that exhaustion is possible, so students can finish and still discuss what went wrong on failed attempts. Early success matters: it shows that a complete tour is a real object, not a legend. From there we enlarge the board carefully, keeping the same questions in play.

A strong sequence asks students to record routes, compare dead ends, and notice recurring traps: isolated corners, poorly timed visits to low-degree squares, and paths that paint themselves into a region with no exit. We introduce language only when it earns its keep. "Fewer exits" is often clearer than "degree" at first; the formal word can wait until students have felt the idea.

In class we use the interactive tour as a shared sandbox, then ask for written traces of promising attempts. Writing a route forces precision. Discussion then separates lucky completion from method: Did you favour edges early? Did you leave an escape? Could you have chosen differently three moves earlier?

For stronger juniors and intermediates, we push towards explanation rather than longer boards alone. Why does this small board work? Where does a near-tour typically fail? Can the same idea reopen a stranded path? The topic stays playful, but the centre of the lesson is controlled search and clear reasoning.