Given N, the number of digits of an integer which is greater than or equal to 2 and a weight W. The task is to find the count of integers that have N digits and weight W.
Note: Weight is defined as the difference between the consecutive digits of an integer.
Examples:
Input : N = 2, W = 3 Output : 6 Input : N = 2, W = 4 Output : 5
In the above example, the total possible 2 digit integers with a weight equal to 3 will be 6. Like the number 14 has weight 3 (4-1) and 25, 36, 47, 58, 69 has weight 3. If we see it carefully we’ll find the logic that if we increment the weight as 5 of a 2-digit number, then the total possible such numbers will be 5. With weight 6 of a 2-digit number, the total possible numbers will be 4 and then 3 and so on. Also, if we increase the number of digits. Say, n equal to 3 with weight 3, then the total possible numbers will be 60 and 600 for n equal to 4 with weight 3 and so on.
Number of digits | Weight —> Total possible such numbers
2|2 —> 7 | 2|3 —> 6 | 2|4 —> 5 | 2|5 —> 4 | 2|6 —> 3 | 2|7 —> 2 | 2|8 —> 1 |
3|2 —> 70 | 3|3 —> 60 | 3|4 —> 50 | 3|5 —> 40 | 3|6 —> 30 | 3|7 —> 20 | 3|8 —> 10 |
4|2 —>700 | 4|3 —>600 | 4|4 —>500 | 4|5 —>400 | 4|6 —>300 | 4|7 —>200 | 4|8 —>100 |
As you can see in the above table that with an increase in the number of digits, the number of numbers with weight ‘w’ is following a pattern, where it is changing in the multiple of 10^(n-2), where ‘n’ is the number of digits.
Below is the step by step algorithm to solve this problem:
- Check if the given Weight(W) is Positive or Negative.
- Subtract Weight(W) from 9 if positive.
- Add Weight to 10 if it is negative and then update the new weight.
- For n digit integer, multiply 10^(n-2) with this updated weight.
- This will give us the number of integers satisfying this weight.
Below is the implementation of above approach:
// CPP program to find total possible numbers // with n digits and weight w #include <iostream> #include<cmath> using namespace std;
// Function to find total possible numbers // with n digits and weight w int findNumbers( int n, int w)
{ int x = 0, sum = 0;
// When Weight of an integer is Positive
if (w >= 0 && w <= 8) {
// Subtract the weight from 9
x = 9 - w;
}
// When weight of an integer is negative
else if (w >= -9 && w <= -1) {
// add the weight to 10 to make it positive
x = 10 + w;
}
sum = pow (10, n - 2);
sum = (x * sum);
return sum;
} // Driver code int main()
{ int n, w;
// number of digits in an
// integer and w as weight
n = 3, w = 4;
// print the total possible numbers
// with n digits and weight w
cout << findNumbers(n, w);;
return 0;
} |
// Java program to find total // possible numbers with n // digits and weight w class GFG
{ // Function to find total // possible numbers with n // digits and weight w static int findNumbers( int n, int w)
{ int x = 0 , sum = 0 ;
// When Weight of an
// integer is Positive
if (w >= 0 && w <= 8 )
{
// Subtract the weight from 9
x = 9 - w;
}
// When weight of an
// integer is negative
else if (w >= - 9 && w <= - 1 )
{
// add the weight to 10
// to make it positive
x = 10 + w;
}
sum = ( int )Math.pow( 10 , n - 2 );
sum = (x * sum);
return sum;
} // Driver code public static void main(String args[])
{ int n, w;
// number of digits in an
// integer and w as weight
n = 3 ;
w = 4 ;
// print the total possible numbers
// with n digits and weight w
System.out.println(findNumbers(n, w));
} } // This code is contributed // by ankita_saini |
# Python3 program to find total possible # numbers with n digits and weight w # Function to find total possible # numbers with n digits and weight w def findNumbers(n, w):
x = 0 ;
sum = 0 ;
# When Weight of an integer
# is Positive
if (w > = 0 and w < = 8 ):
# Subtract the weight from 9
x = 9 - w;
# When weight of an integer
# is negative
elif (w > = - 9 and w < = - 1 ):
# add the weight to 10 to
# make it positive
x = 10 + w;
sum = pow ( 10 , n - 2 );
sum = (x * sum );
return sum ;
# Driver code # number of digits in an # integer and w as weight n = 3 ;
w = 4 ;
# print the total possible numbers # with n digits and weight w print (findNumbers(n, w));
# This code is contributed # by mits |
// C# program to find total possible // numbers with n digits and weight w using System;
class GFG
{ // Function to find total possible // numbers with n digits and weight w static int findNumbers( int n, int w)
{ int x = 0, sum = 0;
// When Weight of an integer
// is Positive
if (w >= 0 && w <= 8)
{
// Subtract the weight from 9
x = 9 - w;
}
// When weight of an
// integer is negative
else if (w >= -9 && w <= -1)
{
// add the weight to 10
// to make it positive
x = 10 + w;
}
sum = ( int )Math.Pow(10, n - 2);
sum = (x * sum);
return sum;
} // Driver code static public void Main ()
{ int n, w;
// number of digits in an
// integer and w as weight
n = 3;
w = 4;
// print the total possible numbers
// with n digits and weight w
Console.WriteLine(findNumbers(n, w));
} } // This code is contributed by jit_t |
<?php // PHP program to find total possible // numbers with n digits and weight w // Function to find total possible // numbers with n digits and weight w function findNumbers( $n , $w )
{ $x = 0; $sum = 0;
// When Weight of an integer
// is Positive
if ( $w >= 0 && $w <= 8)
{
// Subtract the weight from 9
$x = 9 - $w ;
}
// When weight of an integer
// is negative
else if ( $w >= -9 && $w <= -1)
{
// add the weight to 10 to
// make it positive
$x = 10 + $w ;
}
$sum = pow(10, $n - 2);
$sum = ( $x * $sum );
return $sum ;
} // Driver code // number of digits in an // integer and w as weight $n = 3; $w = 4;
// print the total possible numbers // with n digits and weight w echo findNumbers( $n , $w );
// This code is contributed // by Akanksha Rai |
<script> // Javascript program to find total possible
// numbers with n digits and weight w
// Function to find total possible
// numbers with n digits and weight w
function findNumbers(n, w)
{
let x = 0, sum = 0;
// When Weight of an integer
// is Positive
if (w >= 0 && w <= 8)
{
// Subtract the weight from 9
x = 9 - w;
}
// When weight of an
// integer is negative
else if (w >= -9 && w <= -1)
{
// add the weight to 10
// to make it positive
x = 10 + w;
}
sum = Math.pow(10, n - 2);
sum = (x * sum);
return sum;
}
let n, w;
// number of digits in an
// integer and w as weight
n = 3;
w = 4;
// print the total possible numbers
// with n digits and weight w
document.write(findNumbers(n, w));
</script> |
50
Time Complexity: O(log(n))
Auxiliary Space: O(1)