Take three consecutive positive integers and cube them. Add up the digits in each of the three results, and add again until you’ve reached a single digit for each of the three numbers.
For example:
46^3= 97336; 9 + 7 + 3 + 3 + 6 = 28; 2 + 8 = 10; 1 + 0 = 1
47^3= 103823; 1+ 0 + 3 + 8 + 2 + 3 = 17; 1 + 7 = 8
48^3= 110592; 1 + 1 + 0 + 5 + 9 + 2 = 18; 1 + 8 = 9

Arrange the 3 digits in ascending order.

ALWAYS 189 !

Please explain why.

Source: Kendall and Thomas’ Mathematical Puzzles for the Connoisseur (1971)