intertwine, a labyrinthine puzzle tests the sharpest mind. Within this riddle's depths, a story of knights and kings and a treasure untold shall unfold. Imagine a mighty chessboard, with sixty-four squares so grand, where black and white alternate, a captivating land. Upon this board, two knights are placed, noble in their might. Their mission: to find the treasure hidden out of sight. But here's the twist, the tricky part, the puzzle's cunning scheme: the knights must journey together, a duo they must seem. One knight moves north, then two steps to the right, while the other takes a diagonal leap, a path both swift and light. They continue their pursuit, weaving through the chessboard's squares, till they've visited each and every one, proving their thorough care. Now comes the question, the riddle's hidden key: how many times did their paths cross, tell me if you see. Remember, their moves are synchronized, each step taken as a pair. Calculate their crossings, and unravel the secret with care.
Answer: To find the number of times the paths of the two knights cross, we need to analyze their movements on the chessboard. Let's assign coordinates to the squares of the chessboard. We can label the columns as A, B, C, D, E, F, G, and H (from left to right), and the rows as 1, 2, 3, 4, 5, 6, 7, 8 (from bottom to top). Now, let's examine the movements of the knights. The first knight moves one square north and two squares to the right, which can be represented as (2, 1) on the coordinate plane. The second knight takes a diagonal leap, moving one square northeast, which can be represented as (1, 1). We'll start by assuming the initial position of both knights is (0, 0). Now, let's track their movements: The first knight moves to (2, 1). The second knight moves to (1, 1). The first knight moves to (3, 2). The second knight moves to (2, 3). The first knight moves to (4, 4). By analyzing their movements, we can see that the knights' paths intersected once at the coordinate (2, 3). Therefore, the answer is that the paths of the knights cross once.