Question 2

Knights always move from one square color to the opposite color.

Step 1 — Understanding Knight Movement

Let us look at how a knight moves on a chessboard. A knight moves in an 'L' shape. It moves two squares in one direction. Then it moves one square perpendicularly. The possible moves of a knight are shown in the diagram.

This movement has a special property. A knight always changes the square's color. If a knight starts on a white square, it lands on a black square. If it starts on a black square, it lands on a white square. This means a knight always changes color.

Step 2 — Strategy for Peaceful Placement

We want to place 8 knights peacefully. No knight should attack another. We know a knight always moves to the opposite color. So, a knight on a white square attacks only black squares. A knight on a black square attacks only white squares.

Therefore, we can place all knights on squares of the same color. Then no knight can attack another. This is because all potential attack squares are the opposite color. No other knight is on that color.

An 8x8 chessboard has 64 squares in total. Half of these squares are white. Half of these squares are black.

$\text{Number of white squares} = \frac{64}{2}$

$= \mathbf{32}$

$\text{Number of black squares} = \frac{64}{2}$

$= \mathbf{32}$

We can place up to 32 knights peacefully. We do this by putting them all on white squares, or all on black squares.

Step 3 — Placing 8 Knights Peacefully

We need to place only 8 knights. We use the strategy from Step 2. We will place all 8 knights on squares of the same color. Let us choose to place them all on white squares. We can pick any 8 white squares on the board.

For example, we can place the 8 knights on these specific squares: (1,1), (1,3), (1,5), (1,7) (2,2), (2,4), (2,6), (2,8)

These 8 squares are all white squares. We assume (1,1) is a white square. Since all knights are on white squares, any square they could attack would be a black square. No other knight is on a black square. So, no knight attacks another.

Answer

(i) Place all 8 knights on squares of the same color (either all white or all black). (ii) A possible arrangement is to place knights on the white squares at coordinates (1,1), (1,3), (1,5), (1,7), (2,2), (2,4), and (2,6), (2,8).