The problem “Count Odd Numbers in an Interval Range” on Leetcode is an excellent example of how a seemingly simple problem can be interesting from a computational perspective. This article aims to provide a comprehensive guide to the problem, taking you through various approaches to solve it, while analyzing time and space complexities.

## Problem Statement

The problem requires you to count the number of odd numbers between `low`

and `high`

(both inclusive).

#### Example

```
Input: low = 3, high = 7
Output: 3
Explanation: The odd numbers between 3 and 7 are [3,5,7].
```

## The Brute-Force Approach

The first method that comes to mind is to iterate through all the numbers between `low`

and `high`

and count the odd numbers. Although this is not the most efficient solution, it’s a straightforward one.

Python Code:

```
def countOdds(low, high):
count = 0
for num in range(low, high + 1):
if num % 2 != 0:
count += 1
return count
# Test the function
print(countOdds(3, 7)) # Output should be 3
```

### Time Complexity

The time complexity for this approach is O(n), where n is the difference between `high`

and `low`

.

### Space Complexity

The space complexity is O(1), as we are using a constant amount of space.

## An Optimized Approach: Mathematical Calculation

We can solve this problem in a more efficient manner using mathematics. The number of odd numbers between two numbers can be directly calculated if we know the `low`

and `high`

values.

The formula to find the count of odd numbers between `low`

and `high`

can be:

Python Code:

```
import math
def countOdds(low, high):
return math.ceil((high - low) / 2) + (low % 2) + (high % 2) - 1
# Test the function
print(countOdds(3, 7)) # Output should be 3
```

### Time Complexity

The time complexity for this approach is O(1).

### Space Complexity

The space complexity is O(1).

## Edge Cases and Additional Considerations

**Negative Numbers**: The function should also handle scenarios where`low`

and`high`

can be negative. The formula approach will still work.**Single Value Range**: If`low`

and`high`

are the same, the function should return 1 if it’s an odd number or 0 if it’s even.**Zero Values**: The function should return the correct value when`low`

and/or`high`

is zero.

## Conclusion

The “Count Odd Numbers in an Interval Range” problem is an excellent starting point for anyone looking to grasp the importance of optimization in programming. We started with a brute-force approach that works but is not efficient. Then, we leveraged mathematical calculations to derive a formula that allows us to solve the problem in constant time.