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Question 2

Peaceful Knights

Place 8 knights on the chess board so that no knight attacks another. A knight moves in an 'L-shape'. It can move either (a) two steps vertically and one step horizontally, or (b) two steps horizontally and one step vertically. Possible moves of a knight are shown below.

Solution

Understand the Question

A knight moves in an 'L-shape', which means it always moves from a square of one color to a square of the opposite color (white to black or black to white).
Because a knight on a white square can only attack black squares, placing all knights on squares of the same color (all white or all black) guarantees that no knight can attack another.

Step 1 · Property of Knight Movement

A knight moves two squares in one direction and one square perpendicularly (L-shape). Diagram 1

A knight on a white square always lands on a black square, and a knight on a black square always lands on a white square. Therefore, every attack square is of the opposite color.

Step 2 · Color-Based Placement Strategy

Since a knight can only attack squares of the opposite color, placing knights exclusively on squares of the same color ensures no knight attacks another.

On an $8 \times 8$ chessboard with $64$ squares:

\begin{aligned} \text{Number of white squares} &= \dfrac{64}{2} \\[0.6em] &= 32 \end{aligned}

\begin{aligned} \text{Number of black squares} &= \dfrac{64}{2} \\[0.6em] &= 32 \end{aligned}

Thus, up to $32$ non-attacking knights can be placed simultaneously.

Step 3 · Place 8 Knights

Choose any $8$ squares of the same color (for instance, white squares). Diagram 2

A valid placement of $8$ knights on white squares using coordinates $(row, column)$ is:

First row: $(1, 1), (1, 3), (1, 5), (1, 7)$
Second row: $(2, 2), (2, 4), (2, 6), (2, 8)$

Answer

Place all $8$ knights on squares of the same color (e.g., all white squares). A valid configuration is at coordinates $(1, 1), (1, 3), (1, 5), (1, 7), (2, 2), (2, 4), (2, 6), (2, 8)$ .

Common Mistakes

Mixing Colors: Placing knights on both white and black squares without checking L-shaped distances, which easily leads to attacks.
Overcomplicating the Pattern: Trying complex spatial geometric patterns when simply restricting all pieces to one color family trivially solves the problem.

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Peaceful Knights

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