Change one rule — closed tours, fixed endings, wrapped boards

The knight’s tour has a short rule set. The knight moves in an L, visited squares are off limits, and touring every square wins. Add one line and the same board becomes an entirely different puzzle. This game has three such variants.

Closed tours — you must return to the start

After visiting every square, your final square must be one knight-move from where you began, closing the path into a single loop. In graph terms an ordinary tour is a Hamiltonian path and this is a Hamiltonian cycle; the 8×8 chessboard alone has more than 13 trillion closed tours.

What makes it harder is when you have to plan. The end of the journey has to be decided before you set out, because the last square you step on must be a neighbor of the first.

The game uses that property directly. If every neighbor of the starting square has already been visited while unvisited squares remain, the board is finished right there — wherever you end up, you cannot get home. Rather than letting you fill in sixty squares before delivering the bad news, the game says so immediately. Inside the solver the same observation becomes a pruning rule that speeds the search up several times over.

Fixed endings — finish on the flag

If a closed tour is "back to where you started," this one is "over to that spot." A flag sits somewhere on the board and your last move must land on it.

In play it feels quite different. Because you cannot pass through the flagged square, you spend the whole tour avoiding it while making sure it stays reachable at the very end. Step on it early and the board is over — and the game tells you then and there rather than letting you finish the remaining squares for nothing.

Building this chapter required one extra condition: the flagged square must have at least two ways in and out. Choose a square with only one and the puzzle solves itself — such a square can only ever be the end of a path, so the flag demands nothing and becomes decoration. Before that condition existed, a handful of generated boards were exactly that.

Wrapped world — the edges are joined

The left and right edges of the board are connected: leap off the left edge and you come out on the right. Top and bottom remain walls, so the board is neither a plane nor a donut but a cylinder.

This variant changes the terrain rather than the rules. The edge squares gain options, they stop being danger zones, and half the "corners first" heuristic stops applying. Play it on muscle memory and you get trapped somewhere you did not expect.

Candidates for this chapter had to satisfy one more condition: they must be unsolvable without using the wrap. If a board can be finished without ever crossing an edge, the wrap rule does nothing. Proving that absence is where I made a mistake, and two boards shipped in which the rule was meaningless. That story is written up separately.

What one line of rule is worth

None of the three variants changes the board’s shape. Same squares, same knight, same L-shaped move. All that changes is one line in the definition of winning — and that line rewrites the strategy and the difficulty curve completely.

It is far cheaper than making boards bigger. That is the lesson every new chapter keeps teaching.