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

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\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}
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