How many squares are present in the following figure?

Correct — D.
Rule: count squares size by size, then add the separate small squares.

The figure is a 3 × 3 grid with a small square inside the top-left cell and another inside the centre cell.

1 × 1 cells: 9
2 × 2 blocks: 4
3 × 3 outline: 1
Small inner squares: 2

9 + 4 + 1 + 2 = 16 → option (d).

Why the others are wrong

• (a)
  12 — 12 is 9 + 1 + 2. It leaves out the four 2 × 2 squares, each made of four cells, one in each corner of the grid.
• (b)
  18 — 18 counts two squares the figure does not have. Size by size the total is 9 + 4 + 1 + 2 = 16.
• (c)
  14 — 14 is 9 + 4 + 1, the squares of the grid alone. It misses the two small squares drawn inside the top-left and centre cells.

In a grid of n × n cells, squares come in every size from 1 × 1 up to n × n. A 3 × 3 grid holds 9 + 4 + 1 = 14. A square drawn inside a cell adds one each, because it belongs to no larger square.

Count one size at a time and write each count down, so no size gets skipped.

Key facts

• A 3 × 3 grid contains 9 + 4 + 1 = 14 squares.
• Each small square drawn inside a cell adds exactly one.
• In an n × n grid there are (n − k + 1)² squares of side k.

Common traps

• Missing the four 2 × 2 squares, which leads to 12
• Counting the grid alone and forgetting the two small inner squares, which gives 14

No directly related past PYQ was found.