Square Pattern Sequence: Finding the 36-Square Element

The following is a sequence of structures formed by squares with side lengths of 1 cm.

In which element of the sequence are there 36 squares?

To solve this problem, we need to identify in which element a sequence contains exactly 36 squares. If we look closely at the sequence, we notice a structure where each element forms a square with increasing dimensions.

Let's deduce a pattern:

• The 1st element is a $1 \times 1$ square.
• The 2nd element forms a $2 \times 2$ square, consisting of 4 squares.
• The 3rd element forms a $3 \times 3$ square, consisting of 9 squares.
• The 4th element forms a $4 \times 4$ square, consisting of 16 squares.
• The 5th element forms a $5 \times 5$ square, consisting of 25 squares.
• Generalizing this, the $n$-th element forms an $n \times n$ square with $n^2$ squares.

We are seeking for $n$ where $n^2 = 36$. Solving for $n$, we have:

$n^2 = 36$

$n = \sqrt{36}$

$n = 6$

Thus, the 6th element in the sequence is the one that contains 36 squares.

Therefore, the correct answer is $6$.