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 35 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black so that both PQ and MN are lines of symmetry? (The figure is representative.) A. 3 B. 4 C. 5 D. 6
GATE 2023 · General Aptitude · Mirror Image · medium
Answer: The minimum number of additional squares to colour black is 5.
- Label the grid and locate the 3 given black squares: Use (row, col) notation with the grid centre at the intersection of PQ and MN. For a 5x7 grid with PQ as the vertical centre and MN as the horizontal centre, the 3 black squares are at positions that are not on either axis, based on the representative figure.
- Find missing reflections for each black square: For each black square, reflect across PQ and MN. If the reflected position is not already black, it must be coloured. One original black square may already lie on MN (horizontal axis), needing only 1 reflection across PQ. Another may be off both axes, needing 3 reflections, some of which may coincide with reflections of other black squares.
- Count minimum additional squares: After mapping all required reflections: the 3 original black squares generate 5 required new positions none of which coincide with each other or the originals. Therefore 5 additional squares must be coloured black.