How to play the knight's tour
One piece, one rule: you may only move the way a chess knight moves, and every square must be visited exactly once.
How the knight moves
Pick any direction and count two squares. Then turn a right angle and take one more. One, two — turn: that is the whole move. From the middle of the board there are the most of these; near an edge fewer, and from a corner fewest, because the rest of the jumps land off the board.
That is the only move in the game. There is no capture, no other piece and no opponent — the only thing playing against you is the geometry.
The goal
Cover the whole board, standing on each square exactly once. On 6×6 that is thirty-six squares; on 10×10, a hundred. You may never step back onto one you have already used.
It sounds simple until you try it. The difficulty is not finding the next move — there are usually several. It is not cutting off a piece of the board twenty moves before the end, when half the squares are still ahead of you and there is no longer a way into them.
Why the boards start at 6×6
On 4×4 no tour exists at all — that is proven, and no sequence of moves will help. On 5×5 one exists, but the board is small enough to memorise, which turns the puzzle into recall rather than reasoning.
The ceiling is 10×10. A hundred squares is already a long sitting, and on a phone a square under about thirty pixels stops being tappable.
Odd boards: half the squares go dark
A knight always lands on the opposite colour. A tour of n² squares therefore changes colour n²−1 times, which on an even board splits the colours evenly and lets you open anywhere.
On an odd board n² is odd, so the colour you start on gets one square more than the other: (n²+1)/2 against (n²−1)/2. Only the corner colour has that spare square. Opening on the other colour is lost before the first move — the count cannot work out. Rather than let you walk into that silently, the game does not offer those squares.
Three levels of help
Apprentice prints on every reachable square how many moves would be left from it. That is Warnsdorff's rule handed to the player: go where the exit is narrowest, and do not leave narrow places for later.
Regular simply highlights the legal moves. Prodigy shows nothing at all — the board stays quiet and the counting is yours.
Knight's tour · Leap