For a cube root of 1 digit, the maximum number of digits of the cube is 3 digits.
For a cube root of 2 digits, the maximum number of digits of the cube is 6 digits.
For a cube root of 3 digits, the the maximum number of digits of the cube is 9 digits.
The maximum number of digits of a sum of a 9orlessdigit number is 2 digits.
The rate at which the maximum number of digits of a sum of the digits of a number increases is much less than the rate at which the maximum number of digits of a cube increases.
Therefore, the maximum number is where the maximum number of digits is 6 digits, where the maximums are equal, or less.
As the maximum sum of digits of 6digit cubes (cube roots of 47 to 99) is 36 or less, and as 36 is less than 47, the maximum number of digits must then be a 5digit number or less.
The sums of the digits of 5digit cubes (cube roots of 22 to 46) are {9, 10, 17, 18, 19, 26, 27, 28}, therefore, for a 5digit solution, only cubes with sum totals of 26, 27 and 28 are possible.
28
^{3} = 21952. The sum of the digits 2+1+9+5+2 is only 19, thus, the cube root must be less.
27
^{3} = 19683. The sum of the digits 1+9+6+8+3 is 27, which equals the cube root, therefore a 5digit cube is a solution and the 5digit cube is
19683.

Posted by Dej Mar
on 20081010 06:34:33 