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Oo, I stumbled on this: https://buttondown.email/hillelwayne/archive/how-to-memorize-a-larger-multiplication-table/ -- I think (with a little thought) that might be the killer insight into memorising times tables.
For example, for the 47 times table, write down the grid with 7 in the top right:
| 7 4 1
| 8 5 2
| 9 6 3
Then prepend the five times table, knocked down by one for each line you've crossed:
\( \begin{pmatrix} 47 & 94 & 141 \\ 188 & 235 & 282 \\ 329 & 376 & 423 \end{pmatrix} \)
Boom!
[Correction in first para: *7 in the top left.]
I think these two grids are all you need to memorise. Suppose you're multiplying by \(10a+b\), [^0]. Rotate the grid so that \( b\) is in the top left. If \( b \lt 5\), write down the \(a\) times table in front, but bump it up by one every time you cross a vertical line. If \(b\gt5\), write down the \(a+1\) times table, but drop it by one each time you cross a vertical line.
[^0] Sigh, because mathematicians: \(a\) and \(b\) integers, \(0\le a\lt10\) and \(1\le b\le 9\), \(b\ne 5\). Honestly.