Working with Fractions | A

Question 1

Chess Puzzles — Non-attacking Queens

Chess is a popular 2-player strategy game. This game has its origins in India. It is played on an $8 \times 8$ chequered grid. There are 2 sets of pieces — black and white — one set for each player. Find out how each piece should move and the rules of the game.

Here is a famous chess-based puzzle. From its current position, a Queen piece can move along the horizontal, vertical or diagonal. Place 4 Queens such that no 2 queens attack each other. For example, the arrangement below is not valid as the queens are in the line of attack of each other.

Now, place 8 queens on this $8 \times 8$ grid so that no 2 queens attack each other!

Solution

Understand the Question

A queen in chess can move any number of squares horizontally, vertically, or diagonally.
In the $N$ -Queens puzzle, $N$ queens must be placed on an $N \times N$ chessboard such that no two queens attack each other.
This requires every queen to have a unique row, a unique column, and a unique diagonal (i.e. no two queens share the same $\text{row} + \text{column}$ sum or $\text{row} - \text{column}$ difference).

(i) Place 4 Queens on a $4 \times 4$ grid such that no 2 queens attack each other.

Step 1 · Understand Queen Movement and Attack Lines

A queen attacks along horizontal, vertical, and diagonal lines. Two queens cannot share the same row, column, or diagonal. Diagram 1

Step 2 · Place 4 Queens on a $4 \times 4$ Grid

Place one queen in each row and column while avoiding common diagonals. Diagram 2

Queen positions $(\text{row}, \text{column})$ :

\begin{aligned} Q_1 &= (1, 2) \\ Q_2 &= (2, 4) \\ Q_3 &= (3, 1) \\ Q_4 &= (4, 3) \end{aligned}

Check diagonals using $\text{row} + \text{column}$ :

\begin{aligned} 1 + 2 &= 3 \\ 2 + 4 &= 6 \\ 3 + 1 &= 4 \\ 4 + 3 &= 7 \end{aligned}

Check diagonals using $\text{row} - \text{column}$ :

\begin{aligned} 1 - 2 &= -1 \\ 2 - 4 &= -2 \\ 3 - 1 &= 2 \\ 4 - 3 &= 1 \end{aligned}

All sums and differences are distinct, confirming no two queens share a diagonal.

Answer

(i) $(1, 2), (2, 4), (3, 1), (4, 3)$

(ii) Place 8 queens on this $8 \times 8$ grid so that no 2 queens attack each other.

Step 1 · Place 8 Queens on an $8 \times 8$ Grid

Place one queen in each row such that no two queens share the same column or diagonal. Diagram 3

Queen positions $(\text{row}, \text{column})$ :

\begin{aligned} Q_1 &= (1, 5) \\ Q_2 &= (2, 2) \\ Q_3 &= (3, 4) \\ Q_4 &= (4, 7) \\ Q_5 &= (5, 3) \\ Q_6 &= (6, 8) \\ Q_7 &= (7, 6) \\ Q_8 &= (8, 1) \end{aligned}

No two queens share the same row, column, or diagonal.

Answer

(ii) $(1, 5), (2, 2), (3, 4), (4, 7), (5, 3), (6, 8), (7, 6), (8, 1)$

Common Mistakes

Overlooking Diagonal Conflicts: Placing queens in distinct rows and columns while forgetting to verify diagonals (where $\text{row} + \text{col}$ or $\text{row} - \text{col}$ values match).
Row/Column Inversion: Swapping the row and column indices $(\text{row}, \text{column})$ , leading to misplaced pieces.