I suspect he knew it already. It kind of stands out as an almost-repeated digit pattern if you look at a table of cubes (in base 10), as the cube sums concerned are 10³+9³ = 1000+729 and 12³+1³ = 1728+1. Showing that it's the smallest such number is not difficult, but would take a bit of thought to come up with on the spot. You can do it by checking a few combinations of terms from your table:

   n   n³
  -------
   1    1
   2    8
   3   27
   4   64
   5  125
   6  216
   7  343
   8  512
   9  729
  10 1000
  11 1331
  12 1728