Visualizing 2-Turn Knight Moves

While each piece demands study, the knight is a particularly tricky piece to use to its full potential. Its unique movement rules mean it is harder to visualize its path than more straightforward pieces like pawns or bishops. Beginners are often shown the following pattern to help remember the possible moves a knight can make in one turn.

While this is helpful, it does little to help visualize two-turn movement combinations. I decided to look for patterns of two-turn movements so I could analyze plans in-game faster. I considered the following:

• Knights always move to a square of a different color
• Two moves therefore places it back on the same color square it began on

The first pattern describes how the knight can move horizontally or vertically. Since it must end up on the same color square, if the knight stays in the same rank or file, it must move an even number of squares because odd distances change color. Since the knight can move either 1 or 2 squares in a cardinal direction per turn, it can move either 2 or 4 squares directly horizontal or vertical over two turns.

Next, we consider diagonal movement. Squares along the same diagonal all have the same color, so we need to get a bit more involved than with horizontal patterns. By thinking of movement as addition or subtraction from coordinate values, we can use vector math to see how diagonal movement is limited. For example, if a knight moves right 1 square and down 2, we can write that change as (+1, -2). With a second move of 2 left, 1 up (-2, +1), we can add the pairs together for a total change of (-1 , -1), or 1 square left and 1 down from the original position.

Since diagonal movement is defined by an equal change in two dimensions, pairs that have the same absolute value are the possible diagonal moves. Additionally, since knight moves always change 2 spaces (even number) in one dimension and 1 space (odd number) in the other, adding an even and an odd number means the results will always be odd. Thus, over two turns, a knight can move either 1 or 3 squares diagonally.

Next up is a movement pattern that I think of as the "super-knight" move. After two moves, the knight can end up moving 1 square in one direction, and 3 in another. This is helpful because it can help show possible "re-routing" knight plays when a desired landing space is one orthogonal square away. This happens when the direction the knight moves 2 squares in is paired with a move that brings it back the opposite direction by 1 square. The following shows the super-knight move patterns.

Finally, we have moves where the knight moves the same way twice, effectively doubling the movement values for 2 squares in one direction and 4 in another.

Putting all these together shows the expansive options that knights have over two turns, with the white knight signifying 1-move plays and black pawns signifying 2-move plays.

With so many options available, it becomes clear how a shortcut to analyze knight positions can be useful. It is important to remember that each 2 move combo needs a safe landing spot for the knight's first move, but by working backwards from possible end states, it is easier to see which moves you need to look at in the first place.

Cool analysis! You should make a blog about this... or should I?