ISC Practicals and Theory - Dudeney Number

A Dudeney number is a positive whole number that is a perfect cube, and whose digits add up to the cube root of the number. This note gives the program written in the main method alone: it adds the digits, cubes the sum and compares it with the number.

Definition

A number $N$ is a Dudeney number if $N$ is a perfect cube and the sum of the digits of $N$ is equal to the cube root of $N$. In other words, if $s$ is the sum of the digits of $N$, then $s^3 = N$.

• $512$ is a Dudeney number: $512 = 8^3$ and $5 + 1 + 2 = 8$.
• $4913$ is a Dudeney number: $4913 = 17^3$ and $4 + 9 + 1 + 3 = 17$.
• $125$ is not a Dudeney number: $125 = 5^3$, but $1 + 2 + 5 = 8$ and $8^3 = 512$, which is not $125$.
• $1$ is a Dudeney number: $1 = 1^3$ and the sum of its digits is $1$.

Algorithm

1. Accept the number num. If num <= 0, display Invalid input and stop.
2. Copy num into temp and set sum = 0.
3. While temp > 0: take the last digit d = temp % 10, add it to sum and remove it with temp = temp / 10.
4. Find cube = sum * sum * sum. The long type is used so that the cube does not overflow.
5. If cube == num, display that num is a Dudeney number. Otherwise display that it is not.

Program

Everything is written inside main: the input, the loop that adds the digits, the cube and the comparison.

import java.util.Scanner;
public class Dudeney
{
    public static void main(String args[])
    {
        Scanner sc = new Scanner(System.in);
        System.out.print("Enter a number: ");
        int num = sc.nextInt();                 // the number to be checked
        if (num <= 0)
        {
            System.out.println("Invalid input");
        }
        else
        {
            int temp = num;                     // copy of num whose digits are added
            int sum = 0;                        // sum of the digits
            while (temp > 0)
            {
                int d = temp % 10;              // last digit
                sum += d;
                temp /= 10;
            }
            long cube = (long) sum * sum * sum; // cube of the sum of the digits
            if (cube == num)
                System.out.println(num + " is a Dudeney number");
            else
                System.out.println(num + " is not a Dudeney number");
        }
    }
}

Variable description table

Sample input and output

Enter a number: 512
512 is a Dudeney number

Enter a number: 125
125 is not a Dudeney number

Enter a number: 1
1 is a Dudeney number

Enter a number: -5
Invalid input

Dry run

Input: num = 512

First the digits are added, starting with temp = 512 and sum = 0:

The loop stops because temp = 0, and sum = 8.

Then cube = 8 * 8 * 8 = 512. As cube == num is true, the program shows that $512$ is a Dudeney number, as in the first sample run.

Variations

1. Range: "Display all Dudeney numbers between m and n."
2. Recursive technique: "Check whether a number is a Dudeney number, using a recursive method to find the sum of the digits."
3. Perfect cube: "Check whether a number is a perfect cube."
4. Sum of digits: "Find the sum of the digits of a number, using a separate method."