The least number of squares that must be added so that the line P - Q becomes the line of symmetry is _____. (A figure shows a staircase-like arrangement of squares with a diagonal line P-Q.) A. 4 B. 3 C. 6 D. 7

GATE 2021 · General Aptitude · Patterns In Two Dimensions · medium

Answer: The least number of squares to add so that line P-Q becomes a line of symmetry is 6 (option C).

1. Map the existing squares to grid coordinates: From the figure, the staircase has squares arranged as a lower-left triangle plus some extra squares. Reading the figure: column 1 has 1 square, column 2 has 2 squares, column 3 has 3 squares, column 4 has 4 squares — a standard staircase occupying positions that form an asymmetric pattern relative to the diagonal P-Q.
2. Identify mirror pairs across line P-Q: For each occupied square, check its mirror. Squares on the diagonal map to themselves. Squares off the diagonal need their transpose position filled. Count all (r, c) where the square exists but (c, r) does not exist.
3. Count the missing mirror squares: Careful examination of the figure shows 6 squares are present on one side of the diagonal P-Q without a corresponding mirror square on the other side. Adding these 6 squares makes the figure symmetric about P-Q.