## Week 4 Assignment Solutions: Mastering the Joy of Computing using Python

**Description:** This article provides detailed solutions for Week 4 of the NPTEL course "The Joy of Computing using Python." The assignment focuses on concepts such as magic squares, factorial calculations, and understanding Python code snippets. Each question is answered with an explanation to help learners grasp the underlying principles and enhance their coding skills.

**Question 1:**
*Which of the following options provides the general formula for the magic constant of a magic square of size n, where all elements are distinct numbers from 1 to n²?*

**Options:**

- $\frac{n \times (n^2 + 1)}{2}$
- $\frac{n^2 + 2}{2}$
- $\frac{n^2 + n^2}{2}$
- $\frac{n + 1 \times (n^2 + 1)}{2}$

**Answer:** **$\frac{n \times (n^2 + 1)}{2}$**

**Explanation:** The magic constant of a magic square, where all elements are distinct and range from 1 to n², is given by the formula $\frac{n \times (n^2 + 1)}{2}$. This formula ensures that the sum of the numbers in each row, column, and diagonal is the same.

**Question 2:**
*What would the magic constant be for a magic square of size 6, given that all elements in the square are distinct numbers from 1 to 36?*

**Options:**

- 72
- 111
- 90
- 110

**Answer:** **111**

**Explanation:** Using the formula $\frac{n \times (n^2 + 1)}{2}$ with $n = 6$, we calculate the magic constant as $\frac{6 \times (36 + 1)}{2} = \frac{6 \times 37}{2} = \frac{222}{2} = 111$.

**Question 3:**
*Does transposing the magic square give us a new magic square?*

**Options:**

- Yes
- No

**Answer:** **Yes**

**Explanation:** Transposing a magic square, which involves swapping rows with columns, results in a new magic square. This operation does not affect the sum of the numbers in each row, column, or diagonal, thus preserving the properties of the magic square.

**Question 4:**
*Which of the following are valid magic squares?*

**Options:**

- Various matrices provided in the image

**Answer:** **Matrix 3**

**Explanation:** Matrix 3 is a valid magic square as the sum of the elements in each row, column, and diagonal are equal, which is the defining property of a magic square.

**Question 5:**
*What is the minimum number of people required to ensure that at least two of them share the same 30-minute birth interval? The intervals start from 12:00 AM and each interval lasts for half an hour.*

**Options:**

- 25
- 49
- 73
- 100

**Answer:** **49**

**Explanation:** There are 48 half-hour intervals in a day. To ensure that at least two people share the same interval, you need one more than 48, which is 49. This follows the pigeonhole principle, where if you have more "pigeons" than "holes," at least one "hole" must contain more than one "pigeon."

**Question 6:**
*Calculate the magic constant for a 4x4 square, where all elements are distinct numbers from 1 to 16. Is it the same as the magic constant for Ramanujan’s magic square? If yes, enter 0, else enter the absolute difference between the two.*

**Options:**

- Numerical entry

**Answer:** **0**

**Explanation:** The magic constant for a 4x4 square is calculated as $\frac{4 \times (16 + 1)}{2} = 34$. Ramanujan's magic square also has a magic constant of 34, so the difference is 0.

**Question 7:**
*What task does function1() perform?*

**Code:**

`python````
def function1(number):
list1 = []
for i in range(1, number):
if number % i == 0:
list1.append(i)
return list1
```

**Options:**

- Calculate factorial of number n.
- Calculate factors of number n.
- Calculate prime factors of number n.
- Calculate factors of number n excluding n.

**Answer:** **Calculate factors of number n excluding n.**

**Explanation:** The function `function1`

iterates from 1 to `number-1`

and checks for divisibility. If a number divides `number`

with no remainder, it is added to `list1`

. Thus, it returns the factors of `number`

excluding the number itself.

**Question 8:**
*For what n1,n2 flag will the variable flag inside function2() not be equal to True?*

**Options:**

- 2,3
- 0,0
- 1,1
- 1,10

**Answer:** **0,0**

**Explanation:** The flag will remain `False`

when `n1`

and `n2`

are both zero because there will be no iteration over the range, and thus no matching condition to set `flag`

to `True`

.

**Question 9:**
*If all possible pairs of prime numbers between 0 and 20 are given to n1 and n2, for how many pairs would function2 print "Completed"?*

**Options:**

- It will not print "Completed" for any pair.
- It will print "Completed" only for pairs (2,3),(3,5),(5,7), and for the remaining it would not print "Completed".
- It will print "Completed" only for pairs (2,3), and for the remaining other pairs of primes it would not print "Completed".
- It will print "Completed" for all pairs of primes between 0 and 20.

**Answer:** **It will print "Completed" for all pairs of primes between 0 and 20.**

**Explanation:** Function2 checks if the prime numbers are factors of each other, which is true only for pairs of prime numbers. Thus, it will print "Completed" for all pairs.

**Question 10:**
*If numbers or pairs of primes which result in function2 to print "Completed" are greater than 0, can we edit the code in function2() so that "Completed" is never printed for any pair of primes?*

**Options:**

- Yes, we can change the logic for setting flag variable to True.
- Yes, we can change/increase the threshold for the length of list2 in the last if block.
- No, it is logically not possible.
- Yes, we can change the final value of flag to True, instead of False.

**Answer:** **No, it is logically not possible.**

**Explanation:** Given that function2 relies on the logic of prime factorization and checking divisibility, it is not possible to alter the code to avoid printing "Completed" for pairs of prime numbers without fundamentally changing the intended function of the code.

These solutions provide a thorough understanding of the concepts covered in Week 4, helping students grasp the principles of Python programming and problem-solving effectively.