In what sense if this a shortcut, when compared to the normal method for 3-digit or 4-digit squares?

(a+b)^2=a^2+2ab+b^2

Call it **duplex method** or call it **binomial formula**, but really what youâ€™re looking at it this with your example of 807^2:

\begin{array}{rr|ll}
&a &b &\\ \hline
&{\color{gray}{0}}8 &07
\end{array}

I donâ€™t understand what this has to do with â€ś**0 is present**â€ťâ€¦ take 1812^2:

\begin{array}{rr|ll}
&a &b &\\ \hline
&18 &12 \\
\end{array}

\begin{array}{rr|r|r}
&a^2 &2ab &b^2 \\ \hline
&324 &{\color{red}{4}}32 &{\color{red}{1}}44
\end{array}

\begin{array}{rr|r|r}
&{\color{red}{4}} &{\color{red}{1}} & \\ \hdashline
&324 &32 &44 \\ \hline
&\color{blue}{328} &\color{blue}{33} &\color{blue}{44} \\ \hline
\end{array}

The only thing that your **0** does is that you donâ€™t have the carry from the rightmost result, because at most you can get 09^2=81. Is that a shortcut?