IG Math Space

By the end of this class session, students will be able to:

• Understand the concept of a nested loop and how it differs from a simple loop.
• Trace the execution order of nested loops — outer loop vs inner loop.
• Compute the total number of iterations as outer × inner.
• Construct pattern-printing programs using nested loops.
• Perform matrix operations (addition, traversal) using nested loops.
• Use nested loops for 2D data structures such as tables and grids.
• Recognize common errors and choose appropriate loop variable names.

2.1 Definition

A nested loop is a loop placed inside another loop. The inner loop completes all its iterations for each single iteration of the outer loop.

2.2 Why do we need nested loops?

Many real-world problems involve two or more dimensions:

• Rows and columns of a table
• Matrices in mathematics
• Grid-based games (chess, tic-tac-toe)
• Pixel positions on a screen (x, y)
• Star and number patterns

A single loop can only walk a 1D list. For 2D data, we need one loop per dimension — hence a loop inside a loop.

2.3 Syntax

Using for loops:

for outer_var in outer_sequence:
    for inner_var in inner_sequence:
        statement(s)

Using while loops:

while outer_condition:
    while inner_condition:
        statement(s)

Mixed form (a for inside a while, or vice versa) is also valid:

for i in range(3):
    j = 0
    while j < 3:
        print(i, j)
        j += 1

3.1 Execution Order

1. The outer loop starts and picks its first item.
2. Control enters the inner loop, which runs completely.
3. Control returns to the outer loop, which picks its next item.
4. Steps 2–3 repeat until the outer loop finishes.

3.2 Total Number of Iterations

For example, for i in range(3): for j in range(4): executes the inner body 3 × 4 = 12 times.

3.3 Trace Example

Consider the following code:

for i in range(3):
    for j in range(3):
        print(i, j)

Trace:

Conceptual Flowchart — Nested Loops

Fig 1. Flowchart of nested loops — inner loop completes fully for each outer iteration.

3.4 Simple Example — 2D Coordinates

for i in range(3):
    for j in range(3):
        print(i, j)

Explanation: For each value of i (0, 1, 2), the inner loop runs for j = 0, 1, 2. Total iterations = 3 × 3 = 9.

3.5 Nested while Loop Example

i = 1
while i <= 3:
    j = 1
    while j <= 3:
        print(i, "*", j, "=", i * j)
        j += 1
    print("---")
    i += 1

Nested loops are the classic tool for printing patterns. The outer loop controls the number of rows; the inner loop controls what is printed in each row.

Example 1: Right-angled triangle of stars

rows = int(input("Enter number of rows: "))
for i in range(1, rows + 1):
    for j in range(i):
        print("*", end=" ")
    print()

Example 2: Inverted right-angled triangle

rows = int(input("Enter number of rows: "))
for i in range(rows, 0, -1):
    for j in range(i):
        print("*", end=" ")
    print()

Example 3: Pyramid pattern

rows = int(input("Enter number of rows: "))
for i in range(1, rows + 1):
    for j in range(rows - i):
        print(" ", end=" ")
    for k in range(2 * i - 1):
        print("*", end=" ")
    print()

Example 4: Number triangle

rows = int(input("Enter number of rows: "))
for i in range(1, rows + 1):
    for j in range(1, i + 1):
        print(j, end=" ")
    print()

Example 5: Same number in each row

rows = int(input("Enter number of rows: "))
for i in range(1, rows + 1):
    for j in range(1, i + 1):
        print(i, end=" ")
    print()

Example 6: Floyd's Triangle

rows = int(input("Enter number of rows: "))
num = 1
for i in range(1, rows + 1):
    for j in range(1, i + 1):
        print(num, end=" ")
        num += 1
    print()

Example 7: Diamond pattern

rows = int(input("Enter number of rows: "))

# Upper half
for i in range(1, rows + 1):
    for j in range(rows - i):
        print(" ", end="")
    for k in range(2 * i - 1):
        print("*", end="")
    print()

# Lower half
for i in range(rows - 1, 0, -1):
    for j in range(rows - i):
        print(" ", end="")
    for k in range(2 * i - 1):
        print("*", end="")
    print()

Matrices are natural 2D structures. Rows are controlled by the outer loop; columns by the inner loop.

Example 8: Read and print a matrix

rows = int(input("Enter number of rows: "))
cols = int(input("Enter number of columns: "))

matrix = []
for i in range(rows):
    row = []
    for j in range(cols):
        val = int(input(f"Enter element [{i}][{j}]: "))
        row.append(val)
    matrix.append(row)

print("\nThe matrix is:")
for i in range(rows):
    for j in range(cols):
        print(matrix[i][j], end=" ")
    print()

Example 9: Matrix addition (3×3)

A = [[1, 2, 3],
     [4, 5, 6],
     [7, 8, 9]]

B = [[9, 8, 7],
     [6, 5, 4],
     [3, 2, 1]]

C = [[0, 0, 0],
     [0, 0, 0],
     [0, 0, 0]]

for i in range(3):
    for j in range(3):
        C[i][j] = A[i][j] + B[i][j]

print("Sum of matrices:")
for row in C:
    print(row)

Example 10: Matrix multiplication (2×2)

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
C = [[0, 0], [0, 0]]

for i in range(2):
    for j in range(2):
        for k in range(2):
            C[i][j] += A[i][k] * B[k][j]

print("Product of matrices:")
for row in C:
    print(row)

Example 11: Transpose of a matrix

A = [[1, 2, 3], [4, 5, 6]]
T = [[0, 0], [0, 0], [0, 0]]

for i in range(2):
    for j in range(3):
        T[j][i] = A[i][j]

print("Transpose:")
for row in T:
    print(row)

Example 12: Sum of all elements in a matrix

A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
total = 0

for i in range(3):
    for j in range(3):
        total += A[i][j]

print("Sum of all elements:", total)

Example 13: Multiplication tables from 1 to 5

for i in range(1, 6):
    print("Table of", i)
    for j in range(1, 11):
        print(i, "x", j, "=", i * j)
    print()

Example 14: Find all prime numbers up to N

n = int(input("Enter N: "))
print("Prime numbers up to", n, "are:")
for num in range(2, n + 1):
    is_prime = True
    for i in range(2, int(num ** 0.5) + 1):
        if num % i == 0:
            is_prime = False
            break
    if is_prime:
        print(num, end=" ")

Example 15: Print all pairs (i, j) whose sum is a target

numbers = [2, 4, 3, 5, 7, 8, 1]
target = 9

print("Pairs with sum", target, ":")
for i in range(len(numbers)):
    for j in range(i + 1, len(numbers)):
        if numbers[i] + numbers[j] == target:
            print(numbers[i], "+", numbers[j], "=", target)

Example 16: Check whether a matrix is symmetric

A = [[1, 2, 3],
     [2, 5, 6],
     [3, 6, 9]]

symmetric = True
for i in range(3):
    for j in range(3):
        if A[i][j] != A[j][i]:
            symmetric = False
            break

if symmetric:
    print("The matrix is symmetric.")
else:
    print("The matrix is not symmetric.")

1. What is a nested loop? Give a real-world example where nested loops are useful.
2. How many times does the inner loop execute in for i in range(4): for j in range(5):?
3. Explain the execution order of a nested loop with a trace of for i in range(2): for j in range(3): print(i, j).
4. Write a Python program to print a right-angled triangle of stars using nested loops.
5. Write a Python program to print a pyramid pattern of stars.
6. Write a Python program to print Floyd's triangle up to N rows.
7. Write a Python program to add two 3×3 matrices using nested loops.
8. Write a Python program to multiply two 2×2 matrices using nested loops.
9. How can you exit both loops of a nested structure when a condition is met?
10. Explain why break inside a nested loop only terminates the inner loop.