Timothy Browning has discovered that
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 https://gilkalai.wordpress.com/2019/03/09/8866128975287528%C2%B3-8778405442862239%C2%B3-2736111468807040%C2%B3/ (where I found out about this) or https://en.wikipedia.org/wiki/Sums_of_three_cubes (new article I wrote once I found out).
@gnivasch According to https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf this is conjectured to be true
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