**Problem**:

Given a list of integers, return the maximum sum of a pair in the list.

**Example**:

Input: `[4, 2, 6, 2, 13]`

Output: 19

**Explanation**: The maximum pair sum is created by adding the two largest numbers in the list, 13 + 6 = 19.

**Understanding the problem**

This problem is asking you to find the maximum sum of a pair in a list of integers. This means that you need to look at the list of numbers and find the two numbers that give you the largest sum when added together. For example, if your list is **[4, 2, 6, 2, 13]**, the maximum pair sum would be 19 (13 + 6 = 19). You would achieve this result by adding the two largest numbers in the list together.

**Solving this problem**

If you want to solve this problem, you will need to be familiar with manipulating and iterating through lists in Python. You should also have basic knowledge of arithmetic operations such as addition and finding the maximum value of a list. Also, it’s a good idea to practice writing code with efficient time and space complexity.

**Iterating and Modifying Lists in Python**

When manipulating and iterating through a list in Python, you could use a for loop to go through each item in the list or you could use methods such as pop(), append(), or remove() to modify the contents of the list.

– Use methods such as `pop()`, `append()` and `remove()` to modify the contents of a list

**Example** of `pop()`:

```
list = [1, 2, 3, 4]
list.pop(0)
# list is now [2, 3, 4]
```

– Use a `for` loop to iterate through a list

**Example**:

```
list = [1, 2, 3, 4]
for item in list:
print(item)
# 1
# 2
# 3
# 4
```

**Working with Arithmetic operations**

When dealing with arithmetic operations such as addition, you would simply use the ‘+’ operator to add two numbers together. For example, if you wanted to add 4 and 6 together, the result would be 10:

`4 + 6 = 10`

To find the maximum value of a list, you would need to use the built-in Python function max(). This function takes a list as a parameter and returns the maximum value in the list. For example, if you had a list like [3, 8, 2, 1], the maximum value would be 8. This could be done like this:

```
list = [3, 8, 2, 1]
max_val = max(list)
#max_val is now 8
```

**Practicing Code Efficiently**

- Choose efficient algorithms for tasks
- Minimize data storage and operations
- Utilize online platforms to learn complexities

To practice writing code with efficient time and space complexity, you can start by being aware of the type of algorithms you are using. Different types of algorithms take different amounts of resources to execute, so it’s important to choose an efficient algorithm for the task. You should also focus on minimizing the amount of data you need to store in memory and the number of loops or other operations you have to perform.

Practicing code on online platforms such as Leetcode or codingwar can be helpful for familiarizing yourself with different algorithms and their complexities.

## Finding the Maximum Pair Sum in a Python List

To solve this problem, we can loop through the list of integers and find the two largest numbers. This can be done by comparing each integer in the list to the values of two variables, max1 and max2, which store the two largest numbers seen so far.

Here is the code:

```
list = [4, 2, 6, 2, 13]
max1 = list[0]
max2 = list[0]
for num in list:
if num > max1:
max2 = max1
max1 = num
elif num > max2:
max2 = num
pair_sum = max1 + max2
print(pair_sum)
# 19
```

We can initialize these two variables with the first element of the list, which allows us to compare the remaining elements to them. If an element is larger than either max1 or max2, we update one of the variables accordingly.

Once we have reached the end of the list, we know that max1 and max2 contain the two largest numbers in the list, and hence, the maximum pair sum is the sum of these two numbers. This can be done using the ‘+’ operator.

**Time Complexity**

- O(n), or linear time
- Operations performed during each iteration are constant time
- Time complexity directly proportional to size of input

The time complexity of this code is O(n), or linear time. This is because the code iterates through the list once, and the operations performed during each iteration are constant time operations. Therefore, the time complexity is directly proportional to the size of the input and can be written as O(n), where ‘n’ is the size of the list.

**Space Complexity of Code**

- Constant time complexity: O(1)
- No additional data structures used
- Only two variables used to store values

The space complexity of this code is O(1), or constant time. This is because the code does not use any additional data structures and only uses two variables to store values. Therefore, the space complexity is independent of the size of the input and can be written as O(1).

**Final Thoughts**

- Efficient algorithm choice can impact time & space complexity
- Number of variables & data structures matters
- Small details can affect program performance

This challenge is a good example of how small details can have a big impact on the time and space complexity of a program. Choosing an efficient algorithm and keeping track of the number of variables and data structures used can be the difference between an efficient program and a slow program.