Question

A line of symmetry is a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The given figure consists of \(16\) unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both \(PQ\) and \(MN\) form lines of symmetry? (The figure is representative.) \begin{center} \includegraphics[width=0.5\textwidth]{02.jpeg} \end{center}

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

When multiple symmetry lines are required, take one coloured square and generate its full orbit under the reflections. The minimal completion colours exactly the orbit; the rest can remain white.

Answer 1

The Correct Option is C

Step 1: Understand the two symmetries.
- \(PQ\) is the vertical line through the centre of the \(4\times4\) grid, so reflection in \(PQ\) pairs columns \(1\leftrightarrow4\) and \(2\leftrightarrow3\).
- \(MN\) is the diagonal from the bottom–left corner to the top–right corner (the anti-diagonal). Reflection in \(MN\) swaps a square at \((r,c)\) with \((5-c,\,5-r)\).

Step 2: Consequence of having both symmetries.
If a square is coloured, then reflecting it in \(PQ\) must also be coloured; reflecting in \(MN\) must also be coloured. The two reflections together generate a \(90^\circ\) rotation, so a coloured square in a general position forces an orbit of four distinct squares; with both reflections present here, the orbit becomes up to eight positions unless the square lies on an axis (no given black square lies exactly on an axis).

Step 3: Track the orbit of one given black square.
Label rows from top to bottom \(1,2,3,4\) and columns from left to right \(1,2,3,4\). From the picture, the three black squares occupy (for example) \((1,2)\), \((2,1)\), and \((4,3)\). Applying reflections in \(PQ\) and \(MN\) to any one of these produces the same orbit of \(8\) positions: \[ \{(1,2),(1,3),(2,1),(2,4),(3,1),(3,4),(4,2),(4,3)\}. \] Hence, to satisfy both symmetries, all these \(8\) squares must be black.

Step 4: Count additional squares needed.
Already black = \(3\) (given). Total required in the orbit = \(8\). \[ \Rightarrow \text{Additional squares to colour} = 8 - 3 = 5. \]

Final Answer:
\[ \boxed{5} \]