How many shapes?

Example: How Many Squares Are in a 4×4 Grid?

Let’s walk through how to solve one of the most classic shape-counting puzzles:
How many total squares can you find in a 4×4 square grid?

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✅ Step-by-Step Method:

We need to count all square shapes, not just the smallest ones.

A square can be any size: 1×1, 2×2, 3×3, or 4×4.
Let’s count them by size:

🔹 Step 1: Count 1×1 Squares

Each small cell is a 1×1 square.
There are 4 rows and 4 columns:

→ 4 × 4 = 16 squares

🔹 Step 2: Count 2×2 Squares

Each 2×2 square occupies a 2×2 block of cells.
We can fit these starting from rows 1–3 and columns 1–3:

→ (4 − 1) × (4 − 1) = 3 × 3 = 9 squares

🔹 Step 3: Count 3×3 Squares

Each 3×3 square fits within a 3×3 portion of the grid.
They can start from rows 1–2 and columns 1–2:

→ (4 − 2) × (4 − 2) = 2 × 2 = 4 squares

🔹 Step 4: Count 4×4 Squares

There’s only one 4×4 square — the whole grid itself:

→ 1 × 1 = 1 square

✅ Final Count:

16 + 9 + 4 + 1 = 30