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.

While searching for more references for the new Wikipedia article (en.wikipedia.org/wiki/Sums_of_) I found a horrible mistake related to this problem in Wolfram's _A New Kind of Science_, p. 789 (wolframscience.com/nks/p789--i). Wolfram says that the smallest solution to $$x^3+y^3+z^3=2$$ is the known sporadic one,
\begin{align}2&=1214928^3\\ &+ 3480205^3\\ &+ (-3528875)^3. \end{align}
But the parametric solution has many that are smaller: $$(1,1,0)$$, $$(7,-5,-6)$$, $$(49,-47,-24)$$, etc.

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

@11011110 also z is not negative according to Wolfram there. A typo of course but still. A Mastodon instance for maths people. The kind of people who make $$\pi z^2 \times a$$ jokes. Use $$ and $$ for inline LaTeX, and $ and $ for display mode.