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

Let us look closely at the shapes in the "Stacked Squares" section of the diagram.

Step 1 — Counting the squares

We will count the small squares in each shape. The first shape is just one small square. The second shape has two rows and two columns of small squares. The third shape has three rows and three columns of small squares. The fourth shape has four rows and four columns of small squares. The fifth shape has five rows and five columns of small squares.

Number of squares in the 1st shape: $1 \times 1$

$\boxed{1}$

Number of squares in the 2nd shape: $2 \times 2$

$\boxed{4}$

Number of squares in the 3rd shape: $3 \times 3$

$\boxed{9}$

Number of squares in the 4th shape: $4 \times 4$

$\boxed{16}$

Number of squares in the 5th shape: $5 \times 5$

$\boxed{25}$

Diagram 1

Step 2 — Finding the number sequence

We found the number of small squares in each shape. The numbers we got are 1, 4, 9, 16, 25. This sequence is made of numbers that we get by multiplying a whole number by itself. These are called square numbers.

Step 3 — Explaining the reason

Each shape is a big square. This big square is made of smaller squares. If a big square has 'n' rows and 'n' columns of small squares, then it has $n \times n$ small squares. For the first shape, $n = 1$ , so $1 \times 1 = 1$ square. For the second shape, $n = 2$ , so $2 \times 2 = 4$ squares. For the third shape, $n = 3$ , so $3 \times 3 = 9$ squares. For the fourth shape, $n = 4$ , so $4 \times 4 = 16$ squares. For the fifth shape, $n = 5$ , so $5 \times 5 = 25$ squares. So, the number of small squares is always the square of the side length.

Answer

(i) There are 1, 4, 9, 16, 25 little squares, respectively, in each shape of the sequence of stacked squares. (ii) This gives the sequence of square numbers. (iii) Each shape is a large square made from smaller squares. If a large square has 'n' small squares along one side, it will have $n \times n$ total small squares.

Question 23

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