Square of Repdigit Number consisting of Instances of 6

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Theorem

The following pattern holds:

\(\ds 6^2\) \(=\) \(\ds 36\)
\(\ds 3 + 6\) \(=\) \(\ds 9\)


\(\ds 66^2\) \(=\) \(\ds 4356\)
\(\ds 43 + 56\) \(=\) \(\ds 99\)


\(\ds 666^2\) \(=\) \(\ds 443 \, 556\)
\(\ds 443 + 556\) \(=\) \(\ds 999\)


\(\ds 6666^2\) \(=\) \(\ds 44 \, 435 \, 556\)
\(\ds 4443 + 5556\) \(=\) \(\ds 9999\)


\(\ds 66666^2\) \(=\) \(\ds 4 \, 444 \, 355\, 556\)
\(\ds 44 \, 443 + 55 \, 556\) \(=\) \(\ds 99 \, 999\)

and so on.


Proof


This theorem requires a proof.
In particular: Simple but tedious.
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Sources

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