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
M
N P
Q
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.5)**
**1. Governing Concept**:
- **Topic**: Spatial Aptitude & Dual Reflectional Symmetry
- For a $4 \times 4$ grid to possess dual symmetry about both orthogonal bisectors $PQ$ (vertical) and $MN$ (horizontal), any colored square at coordinate $(r, c)$ must have mirrored counterparts at:
$(r, 5-c), \quad (5-r, c), \quad (5-r, 5-c)$
- Evaluating the 3 initial black squares against these required 4-fold reflective orbits reveals that exactly 5 additional squares must be colored black.
**2. Option Analysis**:
- **Option (A) [3] is INCORRECT**: Leaves asymmetric cells along the horizontal axis.
- **Option (B) [4] is INCORRECT**: Leaves one quadrant unmirrored.
- **Option (C) [5] is CORRECT**: Exactly 5 additional squares complete the symmetry orbits across both lines $PQ$ and $MN$.
- **Option (D) [6] is INCORRECT**: Adds redundant non-essential squares.
**3. Final Answer**:
- **Correct Choice**: **C**
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