Smallest N digit number whose sum of square of digits is a Perfect Square

0 0 16 minutes read

Given an integer N, find the smallest N digit number such that the sum of the square of digits (in decimal representation) of the number is also a perfect square. If no such number exists, print -1.
Examples:

Input : N = 2
Output : 34
Explanation:
The smallest possible 2 digit number whose sum of square of digits is a perfect square is 34 because 3² + 4² = 5².
Input : N = 1
Output : 1
Explanation:
The smallest possible 1 digit number is 1 itself.

Method 1:
To solve the problem mentioned above we can use Backtracking. Since we want to find the minimum N digit number satisfying the given condition, the answer will have digits in non-decreasing order. Therefore we generate the possible numbers recursively keeping track of following in each recursive step :

position: the current position of the recursive step i.e. which position digit is being placed.
prev: the previous digit placed because the current digit has to be greater than equal to prev.
sum: the sum of squares of digits placed till now. When digits are placed, this will be used to check whether the sum of squares of all digits placed is a perfect square or not.
A vector which stores what all digits have been placed till this position.

If placing a digit at a position and moving to the next recursive step leads to a possible solution then return 1, else backtrack.
Below is the implementation of the above approach:

C++

// C++ implementation to find Smallest N
// digit number whose sum of square
// of digits is a Perfect Square
 
#include <bits/stdc++.h>
using namespace std;
 
// function to check if
// number is a perfect square
int isSquare(int n)
{
    int k = sqrt(n);
    return (k * k == n);
}
 
// function to calculate the
// smallest N digit number
int calculate(int pos, int prev,
              int sum, vector<int>& v)
{
 
    if (pos == v.size())
        return isSquare(sum);
 
    // place digits greater than equal to prev
    for (int i = prev; i <= 9; i++) {
        v[pos] = i;
        sum += i * i;
 
        // check if placing this digit leads
        // to a solution then return it
        if (calculate(pos + 1, i, sum, v))
            return 1;
 
        // else backtrack
        sum -= i * i;
    }
    return 0;
}
 
string minValue(int n)
{
 
    vector<int> v(n);
    if (calculate(0, 1, 0, v)) {
 
        // create a string representing
        // the N digit number
        string answer = "";
        for (int i = 0; i < v.size(); i++)
 
            answer += char(v[i] + '0');
 
        return answer;
    }
 
    else
        return "-1";
}
 
// driver code
int main()
{
 
    // initialise N
    int N = 2;
 
    cout << minValue(N);
 
    return 0;
}

Java

// Java implementation to find Smallest N
// digit number whose sum of square
// of digits is a Perfect Square
import java.io.*;
import java.util.*;
class GFG{
     
// function to check if
// number is a perfect square
static int isSquare(int n)
{
    int k = (int)Math.sqrt(n);
    return k * k == n ? 1 : 0;
}
 
// Function to calculate the
// smallest N digit number
static int calculate(int pos, int prev,
                     int sum, int[] v)
{
    if (pos == v.length)
        return isSquare(sum);
 
    // Place digits greater than equal to prev
    for(int i = prev; i <= 9; i++) 
    {
        v[pos] = i;
        sum += i * i;
 
        // Check if placing this digit leads
        // to a solution then return it
        if (calculate(pos + 1, i, sum, v) != 0)
            return 1;
 
        // Else backtrack
        sum -= i * i;
    }
    return 0;
}
 
static String minValue(int n)
{
    int[] v = new int[n];
    if (calculate(0, 1, 0, v) != 0)
    {
         
        // Create a string representing
        // the N digit number
        String answer = "";
         
        for(int i = 0; i < v.length; i++)
            answer += (char)(v[i] + '0');
 
        return answer;
    }
    else
        return "-1";
}
 
// Driver code
public static void main(String[] args)
{
 
    // Initialise N
    int N = 2;
 
    System.out.println(minValue(N));
}
}
 
// This code is contributed by jrishabh99

Python3

# Python3 implementation to find Smallest N
# digit number whose sum of square
# of digits is a Perfect Square
from math import sqrt
 
# function to check if
# number is a perfect square
def isSquare(n):
    k = int(sqrt(n))
    return (k * k == n)
 
# function to calculate the
# smallest N digit number
def calculate(pos, prev, sum, v):
 
    if (pos == len(v)):
        return isSquare(sum)
 
    # place digits greater than equal to prev
    for i in range(prev, 9 + 1):
        v[pos] = i
        sum += i * i
 
        # check if placing this digit leads
        # to a solution then return it
        if (calculate(pos + 1, i, sum, v)):
            return 1
 
        # else backtrack
        sum -= i * i
 
    return 0
 
def minValue(n):
    v = [0]*(n)
    if (calculate(0, 1, 0, v)):
 
        # create a representing
        # the N digit number
        answer = ""
        for i in range(len(v)):
 
            answer += chr(v[i] + ord('0'))
 
        return answer
 
    else:
        return "-1"
 
 
# Driver code
if __name__ == '__main__':
 
    # initialise N
    N = 2
 
    print(minValue(N))
 
# This code is contributed by mohit kumar 29

C#

// C# implementation to find Smallest N
// digit number whose sum of square
// of digits is a Perfect Square
using System;
class GFG{
     
// function to check if
// number is a perfect square
static int isSquare(int n)
{
    int k = (int)Math.Sqrt(n);
    return k * k == n ? 1 : 0;
}
 
// Function to calculate the
// smallest N digit number
static int calculate(int pos, int prev,
                     int sum, int[] v)
{
    if (pos == v.Length)
        return isSquare(sum);
 
    // Place digits greater than equal to prev
    for(int i = prev; i <= 9; i++) 
    {
        v[pos] = i;
        sum += i * i;
 
        // Check if placing this digit leads
        // to a solution then return it
        if (calculate(pos + 1, i, sum, v) != 0)
            return 1;
 
        // Else backtrack
        sum -= i * i;
    }
    return 0;
}
 
static string minValue(int n)
{
    int[] v = new int[n];
    if (calculate(0, 1, 0, v) != 0)
    {
         
        // Create a string representing
        // the N digit number
        string answer = "";
         
        for(int i = 0; i < v.Length; i++)
            answer += (char)(v[i] + '0');
 
        return answer;
    }
    else
        return "-1";
}
 
// Driver code
public static void Main()
{
 
    // Initialise N
    int N = 2;
 
    Console.Write(minValue(N));
}
}

Javascript

<script>
 
// Javascript implementation to find Smallest N
// digit number whose sum of square
// of digits is a Perfect Square
 
// function to check if
// number is a perfect square
function isSquare(n)
{
    let k = Math.floor(Math.sqrt(n));
    return k * k == n ? 1 : 0;
}
  
// Function to calculate the
// smallest N digit number
function calculate(pos, prev, sum, v)
{
    if (pos == v.length)
        return isSquare(sum);
  
    // Place digits greater than equal to prev
    for(let i = prev; i <= 9; i++)
    {
        v[pos] = i;
        sum += i * i;
  
        // Check if placing this digit leads
        // to a solution then return it
        if (calculate(pos + 1, i, sum, v) != 0)
            return 1;
  
        // Else backtrack
        sum -= (i * i);
    }
    return 0;
}
  
function minValue(n)
{
    let v = Array.from({length: n}, (_, i) => 0);
    if (calculate(0, 1, 0, v) != 0)
    {
          
        // Create a string representing
        // the N digit number
        let answer = "";
          
        for(let i = 0; i < v.length; i++)
            answer += (v[i]  + 0);
  
        return answer;
    }
    else
        return "-1";
}
  
// Driver Code
     
    // Initialise N
    let N = 2;
    document.write(minValue(N));
  
 // This code is contributed by sanjoy_62.
</script>

Output

Time Complexity: O(sqrt(n))
Auxiliary Space: O(n)

Method 2:
The above-mentioned problem can also be solved using Dynamic Programming. If we observe the question carefully we see that it can be converted to the standard Coin Change problem. Given N as the number of digits, the base answer will be N 1’s, the sum of the square of whose digits will be N.

If N itself is a perfect square then the N times 1 will be the final answer.
Otherwise, we will have to replace some 1’s in the answer with other digits from 2-9. Each replacement in the digit will increase the sum of the square by a certain amount and since 1 can be changed to only 8 other possible digits there are only 8 such possible increments. For example, if 1 is changed to 2, then increment will be 2² – 1² = 3. Similarly, all possible changes are : {3, 8, 15, 24, 35, 48, 63, 80}.

So the problem now can be interpreted as having 8 kinds of coins of the aforementioned values and we can use any coin any number of times to create the required sum. The sum of squares will lie in the range of N (all digits are 1) to 81 * N (all digits are 9). We just have to consider perfect square sums in the range and use the idea of coin change to find the N digits that will be in the answer. One important point we need to take into account is that we have to find the smallest N digit number not the number with the smallest square sum of digits.
Below is the implementation of the above-mentioned approach:

C++

// C++ implementation to find the Smallest
// N digit number whose sum of square
// of digits is a Perfect Square
#include <bits/stdc++.h>
using namespace std;
long long value[8100006];
int first[8100006];
// array for all possible changes
int coins[8] = { 3, 8, 15, 24, 35, 48, 63, 80 };
 
void coinChange()
{
    const long long inf = INT_MAX;
 
    // iterating till 81 * N
    // since N is at max 10^5
    for (int x = 1; x <= 8100005; x++) {
 
        value[x] = inf;
 
        for (auto c : coins) {
            if (x - c >= 0 && value[x - c] + 1 < value[x]) {
                value[x] = min(value[x], value[x - c] + 1);
 
                // least value of coin
                first[x] = c;
            }
        }
    }
}
 
// function to find the
// minimum possible value
string minValue(int n)
{
 
    // applying coin change for all the numbers
    coinChange();
 
    string answer = "";
 
    // check if number is
    // perfect square or not
    if ((sqrt(n) * sqrt(n)) == n) {
        for (int i = 0; i < n; i++)
 
            answer += "1";
 
        return answer;
    }
 
    long long hi = 81 * n;
    long long lo = sqrt(n);
 
    // keeps a check whether
    // number is found or not
    bool found = false;
 
    long long upper = 81 * n;
    long long lower = n;
 
    // sorting suffix strings
    string suffix;
    bool suf_init = false;
 
    while ((lo * lo) <= hi) {
        lo++;
 
        long long curr = lo * lo;
 
        long long change = curr - n;
 
        if (value[change] <= lower) {
 
            // build a suffix string
            found = true;
 
            if (lower > value[change]) {
                // number to be used for updation of lower,
                // first values that will be used
                // to construct the final number later
                lower = value[change];
                upper = change;
                suffix = "";
                suf_init = true;
                int len = change;
 
                while (len > 0) {
                    int k = sqrt(first[len] + 1);
                    suffix = suffix + char(k + 48);
                    len = len - first[len];
                }
            }
 
            else if (lower == value[change]) {
                string tempsuf = "";
                int len = change;
                while (len > 0) {
                    int k = sqrt(first[len] + 1);
                    tempsuf = tempsuf + char(k + 48);
                    len = len - first[len];
                }
 
                if (tempsuf < suffix or suf_init == false) {
                    lower = value[change];
                    upper = change;
                    suffix = tempsuf;
                    suf_init = true;
                }
            }
        }
    }
    // check if number is found
    if (found) {
        // construct the number from first values
        long long x = lower;
        for (int i = 0; i < (n - x); i++)
            answer += "1";
 
        long long temp = upper;
 
        // fill in rest of the digits
        while (temp > 0) {
            int dig = sqrt(first[temp] + 1);
            temp = temp - first[temp];
            answer += char(dig + '0');
        }
        return answer;
    }
    else
        return "-1";
}
 
// driver code
int main()
{
 
    // initialise N
    int N = 2;
 
    cout << minValue(N);
 
    return 0;
}

Java

// Java implementation to find the Smallest
// N digit number whose sum of square
// of digits is a Perfect Square
import java.io.*;
import java.util.*;
 
class GFG {
  static long[] value = new long[(int)8100006];
  static int[] first = new int[8100006];
   
  // array for all possible changes
  static int coins[] = { 3, 8, 15, 24, 35, 48, 63, 80 };
 
  public static void coinChange()
  {
    final long inf = Integer.MAX_VALUE;
 
    // iterating till 81 * N
    // since N is at max 10^5
    for (int x = 1; x <= 8100005; x++) {
 
      value[x] = inf;
 
      for (int c : coins) {
        if (x - c >= 0
            && value[x - c] + 1 < value[x]) {
          value[x] = Math.min(value[x],
                              value[x - c] + 1);
 
          // least value of coin
          first[x] = c;
        }
      }
    }
  }
 
  // function to find the
  // minimum possible value
  public static String minValue(int n)
  {
 
    // applying coin change for all the numbers
    coinChange();
    String answer = "";
 
    // check if number is
    // perfect square or not
    if ((Math.sqrt(n) * Math.sqrt(n)) == n) {
      for (int i = 0; i < n; i++)
 
        answer += "1";
 
      return answer;
    }
 
    long hi = 81 * n;
    long lo = (long)Math.sqrt(n);
 
    // keeps a check whether
    // number is found or not
    boolean found = false;
 
    long upper = 81 * n;
    long lower = n;
 
    // sorting suffix strings
    String suffix = "";
    boolean suf_init = false;
 
    while ((lo * lo) <= hi) {
      lo++;
 
      long curr = lo * lo;
      long change = curr - n;
 
      if (value[(int)change] <= lower) {
 
        // build a suffix string
        found = true;
 
        if (lower > value[(int)change])
        {
           
          // number to be used for updation of
          // lower, first values that will be used
          // to construct the final number later
          lower = value[(int)change];
          upper = change;
          suffix = "";
          suf_init = true;
          int len = (int)change;
 
          while (len > 0) {
            int k = (int)Math.sqrt(first[len]
                                   + 1);
            suffix = suffix + (char)(k + 48);
            len = len - first[len];
          }
        }
 
        else if (lower == value[(int)change]) {
          String tempsuf = "";
          int len = (int)change;
          while (len > 0) {
            int k = (int)Math.sqrt(first[len]
                                   + 1);
            tempsuf = tempsuf + (char)(k + 48);
            len = len - first[len];
          }
 
          if ((tempsuf.compareTo(suffix) < 0)
              || (suf_init == false)) {
            lower = value[(int)change];
            upper = change;
            suffix = tempsuf;
            suf_init = true;
          }
        }
      }
    }
     
    // check if number is found
    if (found) 
    {
       
      // construct the number from first values
      long x = lower;
      for (int i = 0; i < (n - x); i++)
        answer += "1";
 
      long temp = upper;
 
      // fill in rest of the digits
      while (temp > 0) {
        int dig
          = (int)Math.sqrt(first[(int)temp] + 1);
        temp = temp - first[(int)temp];
        answer += (char)(dig + '0');
      }
      return answer;
    }
    else
      return "-1";
  }
 
  // Driver code
  public static void main(String[] args)
  {
    // initialise N
    int N = 2;
 
    System.out.println(minValue(N));
  }
}
 
// This code is contributed by Palak Gupta

Python3

# Python3 implementation to find the Smallest
# N digit number whose sum of square
# of digits is a Perfect Square
value = [0 for _ in range(810006)];
first =  [0 for _ in range(810006)];
 
# array for all possible changes
coins = [ 3, 8, 15, 24, 35, 48, 63, 80 ];
 
def coinChange():
    inf = 99999999;
 
    # iterating till 81 * N
    # since N is at max 10^5
    for x in range(1, 810005 + 1):
 
        value[x] = inf;
 
        for c in coins: 
            if (x - c >= 0 and value[x - c] + 1 < value[x]) :
                value[x] = min(value[x], value[x - c] + 1);
 
                # least value of coin
                first[x] = c;
             
# function to find the
# minimum possible value
def minValue( n):
 
    # applying coin change for all the numbers
    coinChange();
    answer = "";
 
    # check if number is
    # perfect square or not
    if (n == (int(n ** 0.5)) ** 0.5):
         
        for i in range(n):
            answer += "1"
        return answer;
     
    hi = 81 * n;
    lo = int((n ** 0.5));
 
    # keeps a check whether
    # number is found or not
    found = False;
 
    upper = 81 * n;
    lower = n;
 
    # sorting suffix lets
    suffix = "";
    suf_init = False;
 
    while ((lo * lo) <= hi) :
        lo += 1
        curr = lo * lo;
        change = curr - n;
 
        if (value[change] <= lower) :
 
            # build a suffix let
            found = True;
 
            if (lower > value[change]) :
                # number to be used for updation of lower,
                # first values that will be used
                # to construct the final number later
                lower = value[change];
                upper = change;
                suffix = "";
                suf_init = true;
                len1 = change;
 
                while (len1 > 0) :
                    k = int((first[len1] + 1) ** 0.5);
                    suffix = suffix + str(k)
                    len1 = len1 - first[len1];
                 
            elif (lower == value[change]) :
                tempsuf = "";
                len1 = change;
                while (len1 > 0) :
                    k = int((first[len1] + 1) ** 0.5);
                    tempsuf = tempsuf + str(k);
                    len1 = len1 - first[len1];
                 
                if (tempsuf < suffix or suf_init == False): 
                    lower = value[change];
                    upper = change;
                    suffix = tempsuf;
                    suf_init = True;
                 
    # check if number is found
    if (found) :
        # construct the number from first values
        x = lower;
        for i in range(n - x):
            answer += '1'
 
        temp = upper;
         
        # fill in rest of the digits
        while (temp > 0) :
            dig = int((first[temp] + 1) ** 0.5);
            temp = temp - first[temp];
            answer += str(dig);
         
        return answer;
     
    else:
        return "-1";
 
# driver code
 
# initialise N
N = 2;
print(minValue(N));
 
# This code is contributed by phasing17.    

C#

// C# implementation to find the Smallest
// N digit number whose sum of square
// of digits is a Perfect Square
 
using System;
using System.Collections.Generic;
 
class GFG {
  static long[] value = new long[(int)8100006];
  static int[] first = new int[8100006];
   
  // array for all possible changes
  static int[] coins = { 3, 8, 15, 24, 35, 48, 63, 80 };
 
  public static void coinChange()
  {
    long inf = Int32.MaxValue;
 
    // iterating till 81 * N
    // since N is at max 10^5
    for (int x = 1; x <= 8100005; x++) {
 
      value[x] = inf;
 
      foreach (int c in coins) {
        if (x - c >= 0
            && value[x - c] + 1 < value[x]) {
          value[x] = Math.Min(value[x],
                              value[x - c] + 1);
 
          // least value of coin
          first[x] = c;
        }
      }
    }
  }
 
  // function to find the
  // minimum possible value
  public static string minValue(int n)
  {
 
    // applying coin change for all the numbers
    coinChange();
    string answer = "";
 
    // check if number is
    // perfect square or not
    if ((Math.Sqrt(n) * Math.Sqrt(n)) == n) {
      for (int i = 0; i < n; i++)
 
        answer += "1";
 
      return answer;
    }
 
    long hi = 81 * n;
    long lo = (long)Math.Sqrt(n);
 
    // keeps a check whether
    // number is found or not
    bool found = false;
 
    long upper = 81 * n;
    long lower = n;
 
    // sorting suffix strings
    string suffix = "";
    bool suf_init = false;
 
    while ((lo * lo) <= hi) {
      lo++;
 
      long curr = lo * lo;
      long change = curr - n;
 
      if (value[(int)change] <= lower) {
 
        // build a suffix string
        found = true;
 
        if (lower > value[(int)change])
        {
           
          // number to be used for updation of
          // lower, first values that will be used
          // to construct the final number later
          lower = value[(int)change];
          upper = change;
          suffix = "";
          suf_init = true;
          int len = (int)change;
 
          while (len > 0) {
            int k = (int)Math.Sqrt(first[len]
                                   + 1);
            suffix = suffix + (char)(k + 48);
            len = len - first[len];
          }
        }
 
        else if (lower == value[(int)change]) {
          string tempsuf = "";
          int len = (int)change;
          while (len > 0) {
            int k = (int)Math.Sqrt(first[len]
                                   + 1);
            tempsuf = tempsuf + (char)(k + 48);
            len = len - first[len];
          }
 
          if ((tempsuf.CompareTo(suffix) < 0)
              || (suf_init == false)) {
            lower = value[(int)change];
            upper = change;
            suffix = tempsuf;
            suf_init = true;
          }
        }
      }
    }
     
    // check if number is found
    if (found) 
    {
       
      // construct the number from first values
      long x = lower;
      for (int i = 0; i < (n - x); i++)
        answer += "1";
 
      long temp = upper;
 
      // fill in rest of the digits
      while (temp > 0) {
        int dig
          = (int)Math.Sqrt(first[(int)temp] + 1);
        temp = temp - first[(int)temp];
        answer += (char)(dig + '0');
      }
      return answer;
    }
    else
      return "-1";
  }
 
  // Driver code
  public static void Main(string[] args)
  {
    // initialise N
    int N = 2;
 
    Console.WriteLine(minValue(N));
  }
}
 
// This code is contributed by phasing17

Javascript

// JS implementation to find the Smallest
// N digit number whose sum of square
// of digits is a Perfect Square
 
let  value = new Array(8100006).fill(0);
let first = new Array(8100006).fill(0);
 
// array for all possible changes
let coins = [ 3, 8, 15, 24, 35, 48, 63, 80 ];
 
function coinChange()
{
     let inf = 99999999;
 
    // iterating till 81 * N
    // since N is at max 10^5
    for (let x = 1; x <= 8100005; x++) {
 
        value[x] = inf;
 
        for (var c of coins) {
            if (x - c >= 0 && value[x - c] + 1 < value[x]) {
                value[x] = Math.min(value[x], value[x - c] + 1);
 
                // least value of coin
                first[x] = c;
            }
        }
    }
}
 
// function to find the
// minimum possible value
function minValue( n)
{
 
    // applying coin change for all the numbers
    coinChange();
 
    let answer = "";
 
    // check if number is
    // perfect square or not
    if ( Math.floor(Math.sqrt(n)) ** 2 == n) {
    for (let i = 0; i < n; i++)
 
            answer += "1";
 
        return answer;
    }
 
    let hi = 81 * n;
    let lo = Math.floor(Math.sqrt(n));
 
    // keeps a check whether
    // number is found or not
    let found = false;
 
    let upper = 81 * n;
    let lower = n;
 
    // sorting suffix lets
    let suffix;
    let suf_init = false;
 
    while ((lo * lo) <= hi) {
        lo++;
 
        let curr = lo * lo;
 
        let change = curr - n;
 
        if (value[change] <= lower) {
 
            // build a suffix let
            found = true;
 
            if (lower > value[change]) {
                // number to be used for updation of lower,
                // first values that will be used
                // to construct the final number later
                lower = value[change];
                upper = change;
                suffix = "";
                suf_init = true;
                let len = change;
 
                while (len > 0) {
                    let k = Math.floor(Math.sqrt(first[len] + 1));
                    suffix = suffix + (k).toString();
                    len = len - first[len];
                }
            }
 
            else if (lower == value[change]) {
                let tempsuf = "";
                let len = change;
                while (len > 0) {
                    let k = Math.floor(Math.sqrt(first[len] + 1));
                    tempsuf = tempsuf + (k).toString();
                    len = len - first[len];
                }
 
                if (tempsuf < suffix || suf_init == false) {
                    lower = value[change];
                    upper = change;
                    suffix = tempsuf;
                    suf_init = true;
                }
            }
        }
    }
     
    // check if number is found
    if (found) {
        // construct the number from first values
        let x = lower;
        for (let i = 0; i < (n - x); i++)
            answer += "1";
 
        let temp = upper;
         
        // fill in rest of the digits
        while (temp > 0) {
            let dig = Math.floor(Math.sqrt(first[temp] + 1));
            temp = temp - first[temp];
            answer += (dig).toString();
        }
        return answer;
    }
    else
        return "-1";
}
 
 
// driver code
 
// initialise N
let N = 2;
 
console.log(minValue(N));
 
// This code is contributed by phasing17.    

Output:

Time Complexity : O(81 * N)
Auxiliary Space: O(81 * 10⁵)

Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 zambiatek!