A 3 × 3 grid of cells is drawn. In addition, a small square is drawn inside each of the four corner cells and one inside the cent…

Question

A 3 × 3 grid of cells is drawn. In addition, a small square is drawn inside each of the four corner cells and one inside the centre cell. How many squares are present in the following figure?

1. A. 21
2. B. 17
3. C. 23
4. D. 19 (Correct answer)

Correct Answer

Option D — 19

Detailed Solution & Explanation

The correct answer is 19.

Key Points

• Count the squares belonging to the 3 × 3 grid itself, by size:
  • 1 × 1: nine
  • 2 × 2: four (one in each corner position of the grid)
  • 3 × 3: one, the whole grid
  • subtotal = 9 + 4 + 1 = 14
• Add the small squares drawn inside the cells: four in the corner cells and one in the centre = 5.
• Total = 14 + 5 = 19.

Additional Information

• The general result for an n × n grid is the sum of squares 1² + 2² + … + n², i.e. n(n+1)(2n+1)/6. For n = 3 that gives 14 ✓, and for a 4 × 4 grid it gives 30.
• Count by size rather than by position — going size by size is what stops squares being missed or double-counted.
• Squares drawn inside cells are additional figures in their own right; they neither combine with the grid lines nor overlap each other here.
• A companion result worth knowing: the number of rectangles in an n × n grid is [n(n+1)/2]², which for n = 3 is 36 — far more than the 14 squares.