Place N^2 numbers in matrix such that every row has an equal sum

Difficulty Level : Hard
Last Updated : 30 Apr, 2021

Given a number N, place numbers from the range [1, N²] in an NxN matrix such that sum in every row is equal.

Examples:

Input: N = 3
Output: 1 5 9    
        2 6 7    
        3 4 8    
Sum in 1st row: 15
Sum in 2nd row: 15 
Sum in 2nd row: 15 

Input: N = 5
Output: 1 7 13 19 25    
        2 8 14 20 21    
        3 9 15 16 22    
        4 10 11 17 23    
        5 6 12 18 24

A Greedy Approach has been used to fill the matrix, where the insertion of elements in the matrix is done row-wise. The required matrix can be obtained using below steps:

Fill the matrix initially with numbers in the range [1, N²] using matrix traversal.
Traverse the matrix and change every position in a new matrix considering it as answer matrix by answer[i][j] = mat[j][(i+j)%n].

Below is the implementation of the above approach:

C++

// C++ program to distribute n^2 numbers
// to n people
#include <bits/stdc++.h>
using namespace std;
 
vector<vector<int>> solve(vector<vector<int>> arr,
                          int n)
{
     
    // 2D array for storing the final result
    vector<vector<int>> ans(n, vector<int> (n, 0));
 
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
        {
             
            // Using modulo to go to the firs
            // column after the last column
            ans[i][j] = arr[j][(i + j) % n];
        }
    }
    return ans;
}
 
void show(vector<vector<int>> arr, int n)
{
    vector<vector<int>> res = solve(arr, n);
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
        {
            cout << res[i][j] << " ";
        }
        cout << endl;
    }
}
 
// Making a 2D array containing numbers
vector<vector<int>> makeArray(int n)
{
    vector<vector<int>> arr(n, vector<int>(n, 0));
    int c = 1;
    for (int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
        {
            arr[i][j] = c;
            c++;
        }
    }
    return arr;
}
 
// Driver code
int main()
{
    int n = 5;
    vector<vector<int>> arr = makeArray(n);
     
    show(arr, n);
     
    return 0;
}
 
// This code is contributed by divyesh072019

Java

// Java program to distribute n^2 numbers to n people
public class Numbers {
    public static int[][] solve(int[][] arr, int n)
    {
        // 2D array for storing the final result
        int[][] ans = new int[n][n];
 
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                // using modulo to go to the firs
                // column after the last column
                ans[i][j] = arr[j][(i + j) % n];
            }
        }
        return ans;
    }
    public static void show(int[][] arr, int n)
    {
        int[][] res = solve(arr, n);
        for (int i = 0; i < n; i++) {
 
            for (int j = 0; j < n; j++) {
                System.out.print(res[i][j] + " ");
            }
            System.out.println();
        }
    }
    // making a 2D array containing numbers
    public static int[][] makeArray(int n)
    {
        int[][] arr = new int[n][n];
        int c = 1;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++)
                arr[i][j] = c++;
        }
        return arr;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int n = 5;
        int[][] arr = makeArray(n);
        show(arr, n);
    }
}

Python3

# Python3 program to distribute n^2
# numbers to n people
def solve(arr, n):
     
    # 2D array for storing the final result
    ans = [[0 for i in range(n)]
              for j in range(n)]
 
    for i in range(n):
        for j in range(n):
 
            # Using modulo to go to the firs
            # column after the last column
            ans[i][j] = arr[j][(i + j) % n]
 
    return ans
 
def show(arr, n):
     
    res = solve(arr, n)
 
    for i in range(n):
        for j in range(n):
            print(res[i][j], end = " ")
             
        print()
 
# Making a 2D array containing numbers
def makeArray(n):
     
    arr = [[0 for i in range(n)]
              for j in range(n)]
 
    c = 1
 
    for i in range(n):
        for j in range(n):
            arr[i][j] = c
            c += 1
             
    return arr
 
# Driver Code
n = 5
arr = makeArray(n)
 
show(arr, n)
 
# This code is contributed by avanitrachhadiya2155

C#

// C# program to distribute n^2
// numbers to n people
using System;
 
class GFG{
     
static int[,] solve(int[,] arr, int n)
{
     
    // 2D array for storing the final result
    int[,] ans = new int[n, n];
 
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
        {
             
            // Using modulo to go to the firs
            // column after the last column
            ans[i, j] = arr[j, (i + j) % n];
        }
    }
    return ans;
}
 
static void show(int[,] arr, int n)
{
    int[,] res = solve(arr, n);
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
        {
            Console.Write(res[i, j] + " ");
        }
        Console.WriteLine();
    }
}
 
// Making a 2D array containing numbers
static int[,] makeArray(int n)
{
    int[,] arr = new int[n, n];
    int c = 1;
     
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < n; j++)
            arr[i, j] = c++;
    }
    return arr;
}
 
// Driver code
static void Main()
{
    int n = 5;
    int[,] arr = makeArray(n);
     
    show(arr, n);
}
}
 
// This code is contributed by divyeshrabadiya07

Javascript

<script>
 
// Javascript program to distribute n^2 numbers to n people
 
function solve(arr, n)
    {
        // 2D array for storing the final result
        let ans = new Array(n);
         
        // Loop to create 2D array using 1D array
        for (var i = 0; i < ans.length; i++) {
        ans[i] = new Array(2);
        }
  
        for (let i = 0; i < n; i++) {
            for (let j = 0; j < n; j++) {
                // using modulo to go to the firs
                // column after the last column
                ans[i][j] = arr[j][(i + j) % n];
            }
        }
        return ans;
    }
    function show(arr, n)
    {
        let res = solve(arr, n);
        for (let i = 0; i < n; i++) {
  
            for (let j = 0; j < n; j++) {
               document.write(res[i][j] + " ");
            }
            document.write("<br/>");
        }
    }
    // making a 2D array containing numbers
    function makeArray(n)
    {
        let arr = new Array(n);
         
        // Loop to create 2D array using 1D array
        for (var i = 0; i < arr.length; i++) {
        arr[i] = new Array(2);
        }
         
        let c = 1;
        for (let i = 0; i < n; i++) {
            for (let j = 0; j < n; j++)
                arr[i][j] = c++;
        }
        return arr;
    }
     
// driver program   
        let  n = 5;
        let arr = makeArray(n);
        show(arr, n);
  
 // This code is contributed by susmitakundugoaldanga.
</script>

Output:

1 7 13 19 25 
2 8 14 20 21 
3 9 15 16 22 
4 10 11 17 23 
5 6 12 18 24

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