UPSC Prelims 2007 Q143: Science & Technology – Quantitative Aptitude
Six equidistant vertical lines are drawn on a board. Six equidistant horizontal lines are also
drawn on the board cutting the six vertical lines and the distance between any two consecutive
horizontal lines is equal to that between any two consecutive vertical lines. What is the maximum
number of squares thus formed?
- (a) 37
- (b) 55
- (c) 126
- (d) 225
Answer: (b) 55
Explanation
Squares on a grid of 6 × 6 lines
6 vertical and 6 horizontal lines, equally spaced → a 5 × 5 grid of small squares.
Number of squares of each size:
| Size | Count |
|---|---|
| 1 × 1 | 5² = 25 |
| 2 × 2 | 4² = 16 |
| 3 × 3 | 3² = 9 |
| 4 × 4 | 2² = 4 |
| 5 × 5 | 1² = 1 |
| Total | 55 |
Formula: an n × n grid has 1² + 2² + … + n² = n(n+1)(2n+1)/6 squares.
For n = 5: 5 × 6 × 11 / 6 = 55.
Answer: (b) 55.
Why each option is right or wrong
- (a) Incorrect. 37 misses several sizes.
- (b) CORRECT. 25 + 16 + 9 + 4 + 1 = 55.
- (c) Incorrect. 126 is closer to a count of rectangles, not squares.
- (d) Incorrect. 225 = C(6,2)², which is the number of rectangles.
Key point
6 lines each way = 5 × 5 grid → squares = 1² + 2² + 3² + 4² + 5² = 55. Rectangles = C(6,2)² = 225.
Remember
A 5 × 5 grid has 55 squares.
Common mistakes
- Using n = 6 instead of 5 — 6 lines make 5 gaps.
- Counting only the 25 small squares.
- Confusing squares (55) with rectangles (225).
Exam tip
Squares = sum of squares (Σk²). Rectangles = C(lines, 2) × C(lines, 2).
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)