Timothy Browning has discovered that
\begin{align} 33&=8866128975287528^3\\ &+(-8778405442862239)^3\\ &+(-2736111468807040)^3. \end{align}

This settles all but one case of which two-digit numbers can be represented as a sum of three cubes. The remaining case is $n=42$.

For more, see gilkalai.wordpress.com/2019/03 (where I found out about this) or en.wikipedia.org/wiki/Sums_of_ (new article I wrote once I found out).

Update: Kalai now says that Browning now says that the discovery was by Andrew Booker.

@11011110 I wonder if every such number can be written as the sum of three cubes in *infinitely many ways*.

@gnivasch According to people.maths.bris.ac.uk/~maarb this is conjectured to be true

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