Proof without words: $1+8\times\text{triangular number}$ is an odd perfect square

Question

A recent question asked how to show that $8T_n+1$ is a perfect square if $T_n$ is a triangular number. This follows immediately from $T_n=\frac12 n(n+1)\implies 8T_n+1=4n^2+4n+1=(2n+1)^2$.

Can this be proven without words?

Answer 1

The answer turns out to be quite simple, as @JMoravitz notes in his comment. In fact, Mathworld's page on triangular numbers includes precisely the right image:

Answer 2

$$ \begin{matrix} \color{red}{\bullet} & \color{red}{\bullet} & \color{red}{\bullet} & \color{red}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{green}{\bullet} \\ \color{orange}{\bullet} & \color{red}{\bullet} & \color{red}{\bullet} & \color{red}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} \\ \color{orange}{\bullet} & \color{orange}{\bullet} & \color{red}{\bullet} & \color{red}{\bullet} & \color{blue}{\bullet} & \color{blue}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} \\ \color{orange}{\bullet} & \color{orange}{\bullet} & \color{orange}{\bullet} & \color{red}{\bullet} & \color{blue}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} & \color{green}{\bullet} \\ \color{orange}{\bullet} & \color{orange}{\bullet} & \color{orange}{\bullet} & \color{orange}{\bullet} & \color{black}{\circ} & \color{purple}{\bullet} & \color{purple}{\bullet} & \color{purple}{\bullet} & \color{purple}{\bullet} \\ \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{magenta}{\bullet} & \color{yellow}{\bullet} & \color{purple}{\bullet} & \color{purple}{\bullet} & \color{purple}{\bullet} \\ \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} & \color{purple}{\bullet} & \color{purple}{\bullet} \\ \color{cyan}{\bullet} & \color{cyan}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} & \color{purple}{\bullet} \\ \color{cyan}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{magenta}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} & \color{yellow}{\bullet} \end{matrix} $$

Answer 3

And you can convert the proof without words back into algebra (which has the advantage of showing that it is true for any $n$):

$\begin{array}\\ 8T_n &=4(2T_n)\\ &=4(2(\frac{n(n+1}{2}))\\ &=4(n(n+1))\\ &=4n^2+4n\\ &=4n^2+4n+1-1\\ &=(2n+1)^2-1\\ \end{array} $