What are the two consecutive triangular numbers that sum up to the 10th square number? How can this relationship be visualized?

A square number $S_n$ can be formed by adding two consecutive triangular numbers, $T_n$ and $T_{n-1}$. The formula for the $n$-th triangular number is $T_n = \frac{n(n+1)}{2}$.

The $10^{th}$ square number is $S_{10} = 10^2 = 100$.
According to the relationship, $S_{10} = T_{10} + T_{10-1} = T_{10} + T_9$.

To check: $T_9 + T_{10} = 45 + 55 = 100$, which is $10^2$.

The triangular numbers used to make the $10^{th}$ square number are the $9^{th}$ and $10^{th}$ triangular numbers.

The $9^{th}$ triangular number is 45.
The $10^{th}$ triangular number is 55.

To draw the pattern for the $10^{th}$ square number (100 dots):

• Draw the $10^{th}$ triangular number ($T_{10}$) as a right-angled triangle of dots. It will have 10 dots in the bottom row, 9 in the row above it, and so on, up to 1 dot at the top.
• Draw the $9^{th}$ triangular number ($T_9$) as another right-angled triangle of dots. It will have 9 dots in the bottom row, 8 in the row above it, and so on, up to 1 dot at the top.
• Rotate the $T_9$ pattern by $180^\circ$ and place it next to the $T_{10}$ pattern.
• When combined, the two triangular patterns will perfectly form a $10 \times 10$ square grid of dots.

The triangular numbers used are $\text{45 and 55}$.