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Another puzzle from Kendall and Thomas’ Mathematical Puzzles for the Connoisseur (1971):

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:

463 = 97336; 9 + 7 + 3 + 3 + 6 = 28; 2 + 8 = 10; 1 + 0 = 1
473 = 103823; 1 + 0 + 3 + 8 + 2 + 3 = 17; 1 + 7 = 8
483 = 110592; 1 + 1 + 0 + 5 + 9 + 2 = 18; 1 + 8 = 9

Putting the three digits in ascending order will always give the result 189. Why?

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