To understand where a knight can move in a chess game, we need to know the initial setup. A chess board is an 8 8 grid. To be able to refer to the individual positions on the board, we will place the board so that its lower left corner is at the origin, make each square in the grid have side length 1, and label each square with the point at the lower left corner. This is illustrated at left in Figure 4.5. Figure 4.5: Initial knight placement and moves. Each player has two knights to start the game, for one player the knights would begin in positions (1, 0) and (6, 0). Because of the symmetry of the knights moves, we will only analyze the moves of the knight that begins at position (1, 0). This knight has only three allowable moves from its starting point (assuming that the board is empty), as shown at right in Figure 4.5. The questions we will ask are: given any position on the board, can the knight move from its start position to that position using only knight moves and, what sequence of moves will make that happen. To answer these questions we will use linear combinations of knight moves described as vectors. Each knight move can be described by a vector. A move one position to the right and two up can be represented at n₁= . Three other moves are n₂ = , n₃ = , and n₄ = . The other four knight moves are the opposites of these four. Any sequence of moves by the knight is given by the linear 96 Section 4. Vector Representation combination x1n1 + x2n2 + x3n3 + x4n4. A word of caution: the knight can only make complete moves, so we are restricted to integer (either positive or negative) values for x1, x2, x3, and x4. You can use the GeoGebra app at https://www.geogebra.org/m/dfwtskrj to see the affects the weights have on the knight moves. We should note here that since addition of vectors is commutative, the order in which we apply our moves does not matter. However, we may need to be careful with the order so that our knight does not leave the chess board. Project Activity 4.1. (a) Explain why the vector equation 1 0 + x1n1 + x2n2 + x3n3 + x4n4 = will tell us if it is possible for the knight to move from its initial position at (1, 0) to the position (5,2). (b) Find all solutions, if any, to the system from part (a). If a solution exists, find weights x1, x2, x3, and x4 to accomplish this move. Is there more than one sequence of possible moves? (Hint: Be careful we must have solutions in which x1, x2, x3, and x4 are integers.) You can check your solution with the GeoGebra app at https://www.geogebra.org/ m/dfwtskrj. Project Activity 4.1 shows that it is possible for our knight to move to position (5, 2) on the board. We would like to know if it is possible to move to any position on the board. That is, we would like to know if the span of the four moves n1, n2, n3, and n4 will allow our knight to cover the entire board. This takes a bit more work