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The Knight's Tour is a genuine classical problem: move a chess knight so that it visits every square on the board exactly once. Euler wrote about it in the eighteenth century. It looks impossible until you learn the one heuristic that makes it tractable, at which point it becomes surprisingly reliable.
The knight starts on a square you choose.
Move it in the standard L-shape: two squares one way, one the other.
Each square may be visited only once.
Visit every square on the board to complete the tour.
Larger boards unlock as you succeed.
Look-ahead and constraint management. The classic solution heuristic is worth learning on its own merits — it is a clean example of how a simple local rule can solve a problem that brute force cannot.
Use Warnsdorff's rule: always move to the square that has the fewest onward moves available. It is the standard heuristic and it works remarkably well.
Do the corners early. Corner squares have the fewest entrances and are what strand you if left for last.
Work the edges before the centre. Centre squares stay reachable for much longer.
If two squares tie on onward moves, take the one closer to an edge.
On a standard 8×8 board, yes — many tours exist. Some smaller boards have none at all, which is a genuinely interesting result.
Move to the square with the fewest onward options. It was proposed in 1823 and it solves the ordinary board reliably, though it is not guaranteed in general.
Only the knight's move. No other chess knowledge is involved.
Because corners have only two entrances. Once both are used, that square is unreachable. Visit them early.