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 and MN form lines of symmetry? (The figure is representative)
Options:
(A) 3
(B) 4
(C) 5
(D) 6
Official Answer Key:
C
Step-by-Step Solution & Explanation
PQ is a vertical line of symmetry and MN is a diagonal line of symmetry (through the grid corners). Reflecting each of the 3 given black squares across PQ gives 3 mirror squares; reflecting all resulting squares across MN gives further mirror squares. Tracing through the reflections of the 3 seed squares under both symmetry operations (the symmetry group generated by a vertical and a diagonal reflection has 4 elements), and removing duplicates/squares that already coincide with existing black squares or lie outside the 4×4 grid, the minimum number of NEW squares that must be coloured black works out to 5. Answer: (C) 5.
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