A line of symmetry is defined as 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 \(PQ\) (vertical) and \(MN\) (the bottom-left to top-right diagonal) form lines of symmetry? (The figure is representative)
Show Hint
When multiple symmetries are required, take the \textbf{orbit/closure} of given cells under all reflections—keep adding the reflected cells until no new ones appear.
For a \(4\times4\) grid, reflection over the mid-vertical is \( (r,c)\!\mapsto\!(r,5-c) \) and over the anti-diagonal is \( (r,c)\!\mapsto\!(5-c,5-r) \).
Step 1: Model the \(4\times4\) grid with coordinates \((r,c)\) where \(r=1\) at the top row and \(c=1\) at the left column. From the figure, the initially black squares are \((1,3), (2,1), (4,3)\). Step 2: Enforce symmetry about the vertical line \(PQ\) (the line between columns 2 and 3). Reflection in \(PQ\) maps \((r,c)\mapsto (r,5-c)\). Therefore the current blacks force \((1,2), (2,4), (4,2)\) to be black. Step 3: Enforce symmetry about the diagonal line \(MN\) (from bottom-left to top-right). Reflection in \(MN\) maps \((r,c)\mapsto (5-c,\,5-r)\). Applying this to all black squares obtained so far yields the additional required squares \((3,1)\) and \((3,4)\). Step 4: The closure under both reflections is
\[
\{(1,2),(1,3),(2,1),(2,4),(3,1),(3,4),(4,2),(4,3)\},
\]
which contains \(8\) black squares in total. Since \(3\) were already black, the minimum number of additional squares required is \(8-3=5\). \(\boxed{5}\)
