How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?

Question

Homi · Accepted Answer

Understand the Question

The sequence of "Stacked Squares" consists of square grids of increasing side lengths: $1 \times 1$ , $2 \times 2$ , $3 \times 3$ , $4 \times 4$ , and $5 \times 5$ .
To find the total number of small squares in each grid, multiply the number of rows by the number of columns ( $n \times n = n^2$ ).
This creates the sequence of square numbers (or perfect squares).

(i) How many little squares are there in each shape of the sequence of Stacked Squares?

Step 1 · Count the Small Squares in Each Shape

Diagram 1

Calculate the number of unit squares in each shape:

Answer

(i) $1, 4, 9, 16, 25$

(ii) Which number sequence does this give?

Step 1 · Identify the Number Sequence

The numbers representing the count of small squares are: $1, 4, 9, 16, 25, \dots$

Since each term is obtained by multiplying a positive integer by itself ( $n \times n$ ), this gives the sequence of square numbers.

Answer

(ii) Sequence of square numbers ( $1, 4, 9, 16, 25, \dots$ )

(iii) Can you explain why?

Step 1 · Explain the Reason

Each shape is a large square formed by a grid of smaller unit squares with $n$ rows and $n$ columns.

Total number of small squares: $\text{Total squares} = n \times n = n^2$

Therefore, the number of small squares is always the square of the side length $n$ .

Answer

(iii) Each shape is an $n \times n$ grid of small squares, giving a total of $n \times n = n^2$ little squares.

Common Mistakes

Perimeter vs Area: Confusing the total number of unit squares inside the shape ( $n^2$ ) with the number of unit squares along its boundary.
Linear Scaling vs Squaring: Thinking the count increases by a constant multiple like $4n$ rather than multiplying side by side ( $n \times n$ ).

Question 23