Square Pattern Sequence: Finding the 64-Square Structure Position

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

Is it possible to have a structure in the series that has 64 squares? If so, what element of the series is it?

To determine the sequence that forms these structures, observe the pattern that these are square grids building up: 1-by-1, 2-by-2, 3-by-3. This creates the series $1^2, 2^2, 3^2, \ldots, n^2$. We are asked to verify if 64 can be represented in this series.

Let's solve $n^2 = 64$ to find out which structure provides 64 squares:

• Start by taking the square root of both sides: $\sqrt{n^2} = \sqrt{64}$.
• This simplifies to $n = 8$, since $\sqrt{64} = 8$.

This confirms that the structure with 64 squares is indeed possible and corresponds to the 8th element in our series.

Therefore, the element of the series with 64 squares is $8$.