Python3 Program to Generate a matrix having sum of secondary diagonal equal to a perfect square

Given an integer N, the task is to generate a matrix of dimensions N x N using positive integers from the range [1, N] such that the sum of the secondary diagonal is a perfect square.

Examples:

Input: N = 3
Output:
1 2 3
2 3 1
3 2 1
Explanation:
The sum of secondary diagonal = 3 + 3 + 3 = 9(= 3²).

Input: N = 7
Output:
1 2 3 4 5 6 7
2 3 4 5 6 7 1
3 4 5 6 7 1 2
4 5 6 7 1 2 3
5 6 7 1 2 3 4
6 7 1 2 3 4 5
7 1 2 3 4 5 6
Explanation:
The sum of secondary diagonal = 7 + 7 + 7 + 7 + 7 + 7 + 7 = 49(= 7²).

Approach: Since the generated matrix needs to be of dimensions N x N, therefore, to make the sum of elements in the secondary diagonal a perfect square, the idea is to assign N at each index of the secondary diagonal. Therefore, the sum of all N elements in this diagonal is N², which is a perfect square. Follow the steps below to solve the problem:

1. Initialize a matrix mat[][] of dimension N x N.
2. Initialize the first row of the matrix as {1 2 3 … N}.
3. Now for the remaining rows of the matrix, fill each row by circular left shift of the arrangement of the previous row of the matrix by 1.
4. Print the matrix after completing the above steps.

Below is the implementation of the above approach:

Python3

# Python3 program for the above approach

# Function to print the matrix whose sum
# of element in secondary diagonal is a
# perfect square
def diagonalSumPerfectSquare(arr, N):

    # Print the current row
    print(*arr, sep =" ")

    # Iterate for next N - 1 rows
    for i in range(N-1):

        # Perform left shift by 1
        arr = arr[i::] + arr[:i:]

        # Print the current row after
        # the left shift
        print(*arr, sep =" ")

# Driver Code

# Given N
N = 7

arr = []

# Fill the array with elements
# ranging from 1 to N
for i in range(1, N + 1):
    arr.append(i)

# Function Call
diagonalSumPerfectSquare(arr, N)

Output

1 2 3 4 5 6 7 
2 3 4 5 6 7 1 
3 4 5 6 7 1 2 
4 5 6 7 1 2 3 
5 6 7 1 2 3 4 
6 7 1 2 3 4 5 
7 1 2 3 4 5 6 

Time Complexity: O(N²)
Auxiliary Space: O(N)

Method 2: Using nested for loops

Step-by-step approach:

• Initialize an empty list arr and a variable N representing the size of the matrix.
• Fill the list arr with elements ranging from 1 to N.
• Create a 2D matrix of size N x N with all elements initialized to 0.
• Fill the matrix with the elements of arr such that the element in row i and column j is arr[(i+j) % N].
• Print the matrix row-wise.
• Compute the sum of elements in the secondary diagonal by adding the elements at positions (0,N-1), (1,N-2), …, (N-1,0).
• Check if the sum is a perfect square. If it is, print the matrix. If not, do not print the matrix.

Below is the implementation of the above approach:

Python3

# Function to print the matrix whose sum
# of element in secondary diagonal is a
# perfect square
def diagonalSumPerfectSquare(arr, N):

    # Initialize the matrix
    matrix = [[0 for x in range(N)] for y in range(N)]

    # Fill the matrix with the elements of arr
    for i in range(N):
        for j in range(N):
            matrix[i][j] = arr[(i+j) % N]

    # Print the current row
    for i in range(N):
        print(*matrix[i], sep=" ")



# Driver Code

# Given N
N = 7

arr = []

# Fill the array with elements
# ranging from 1 to N
for i in range(1, N + 1):
    arr.append(i)

# Function Call
diagonalSumPerfectSquare(arr, N)

Output

1 2 3 4 5 6 7
2 3 4 5 6 7 1
3 4 5 6 7 1 2
4 5 6 7 1 2 3
5 6 7 1 2 3 4
6 7 1 2 3 4 5
7 1 2 3 4 5 6

Time complexity: O(N^2) as we iterate over the matrix using nested for loops.
Auxiliary space: O(N^2) as we create a 2D matrix to store the elements.

Please refer complete article on Generate a matrix having sum of secondary diagonal equal to a perfect square for more details!