Geometric Square Sequence: Finding the 46-Square Structure Pattern

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

Is it possible for a structure to have 46 squares? If so, which element of the series is it?

To determine whether a structure can have exactly 46 squares and identify which element of the series it corresponds to, we must first recognize the growth pattern of the series of squares in the diagram.

From examining patterns in geometric series, commonly encountered shapes include arrangements forming triangles or squares. Typically, triangular numbers are related to sums of consecutive integers:

The $n$-th triangular number is given by:

Let's calculate the first few triangular numbers to understand the sequence:

• $T_1 = \frac{1 \times (1+1)}{2} = 1$
• $T_2 = \frac{2 \times (2+1)}{2} = 3$
• $T_3 = \frac{3 \times (3+1)}{2} = 6$
• $T_4 = \frac{4 \times (4+1)}{2} = 10$
• $T_5 = \frac{5 \times (5+1)}{2} = 15$
• $T_6 = \frac{6 \times (6+1)}{2} = 21$
• $T_7 = \frac{7 \times (7+1)}{2} = 28$
• $T_8 = \frac{8 \times (8+1)}{2} = 36$
• $T_9 = \frac{9 \times (9+1)}{2} = 45$
• $T_{10} = \frac{10 \times (10+1)}{2} = 55$

Now, check if 46 is among these numbers, as this would indicate a structure with exactly that many squares.

Checking the sequence above, $46$ does not appear. $45$ is the highest number before $46$, and $55$ is the nearest higher one.

Since $46$ is not found in the series of triangular numbers, it is not possible for a structure in this series to have exactly 46 squares.

Therefore, the final answer is No.