The sum of the squares of 3 natural numbers is 1029, and they are in the proportion 1 : 2 : 4. The difference between greatest number and smallest number is:

21

The proportion of the numbers = 1 : 2 : 4 

Numbers can be represented by x, 2x, 4x

Sum of squares of these numbers = x² + 4x² + 16x² = 1029(given)

21x² = 1029

    x² = 49   or x = 7

Thus, the greatest number = 4 x 7 = 28

and the smallest number = 7

Their difference = 21 - 7 =21