I have list of number that i want to sort based on the most significant digit nut the whole number as a whole 
For example 9 is bigger then 34 because the most siginifcate digite here 9 > 34 
Malgre que 34 is bigger then 9 as number 

Q- what the maxthematic trick to do it ? 

To sort a list of numbers based on their most significant digit (the leftmost digit), you can use a mathematical method to extract that digit from each number and then sort according to these extracted digits.

### Mathematical Trick to Extract the Most Significant Digit:

For any positive integer \( n \), the most significant digit \( D \) can be obtained using the formula:

\[
D = \left\lfloor \dfrac{n}{10^{\left\lfloor \log_{10} n \right\rfloor}} \right\rfloor
\]

### Explanation:

1. **Calculate the Base-10 Logarithm**:
   - Compute \( \log_{10} n \), which gives the exponent where the number \( n \) fits on the power of 10 scale.
   - Example: \( \log_{10} 34 \approx 1.531 \).

2. **Find the Exponent**:
   - Take the floor of the logarithm to get the highest power of 10 less than or equal to \( n \).
   - Example: \( \left\lfloor \log_{10} 34 \right\rfloor = 1 \).

3. **Compute 10 Raised to That Exponent**:
   - Calculate \( 10^{\left\lfloor \log_{10} n \right\rfloor} \).
   - For \( n = 34 \), this is \( 10^1 = 10 \).

4. **Divide and Floor**:
   - Divide \( n \) by \( 10^{\left\lfloor \log_{10} n \right\rfloor} \) and take the floor to get the most significant digit.
   - Example: \( D = \left\lfloor \dfrac{34}{10} \right\rfloor = 3 \).

### Implementing the Sorting:

1. **Extract the Most Significant Digit for Each Number**:
   - Apply the formula above to each number in your list.

2. **Sort the Numbers Based on the Extracted Digit**:
   - Use the extracted digits as sorting keys.
   - If two numbers share the same most significant digit, you can choose to either:
     - Sort them based on the next significant digit.
     - Maintain their original order.

### Example:

Given the list: \([9, 34, 56, 7, 81]\)

- Extract the most significant digits:
  - 9 → 9
  - 34 → 3
  - 56 → 5
  - 7 → 7
  - 81 → 8

- Sort based on these digits:
  - Highest to lowest: \(9, 8, 7, 5, 3\)
  - Ordered list: \([9, 81, 7, 56, 34]\)

### Notes:

- For single-digit numbers (1–9), the number itself is the most significant digit.
- This method works for any positive integer.
- Handle numbers with the same most significant digit based on specific needs, such as considering the next significant digit.

### Conclusion:

To sort numbers by their most significant digit, extract that digit using the formula:

\[
D = \left\lfloor \dfrac{n}{10^{\left\lfloor \log_{10} n \right\rfloor}} \right\rfloor
\]

Then, sort the numbers based on these digits.


Q- oky but these one edge case we need to check it ? 
If both share the same signinficate digite we need to move to the next significant digit and sol one .. but how to do it  mathematically and algorithmicly  ? 


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This leetCode problem is a direct application of the previous idea oky 

179. Largest Number
Difficulty: Medium
Problem Statement:
Given a list of non-negative integers nums, arrange them such that they form the largest number and return it as a string (due to potential size constraints).
Example 1:
Input: nums = [10, 2]
Output: "210"
Example 2:
Input: nums = [3, 30, 34, 5, 9]
Output: "9534330"
Constraints:
1 <= nums.length <= 100
0 <= nums[i] <= 10^9


Q- can you explain it using you own words ? 
So we are given a list of integers , we need to find the best arrangement possible to construct the biggest element possible

Q- what is the input/output format ? 

Input : List[int] 
Output : string 

Q- can you go through an example ? 

Input: nums = [3, 30, 34, 5, 9]
Output: "9534330"


Here we have 3 , 30 ,34 , 5 , 9 
The biggest one is “9534330” 

But how i think we need to build a custom sorting algorithm that sort the input array not based on the actual value of the element but the based on the most significate one possible 
→ one the mean not based of the actual but bazed on the most significate element it mean in this example 9 is bigger then 34 “malgre que” 34 is actually bigger then 9 

Levraging the previous mathematical trick in the previous article and using functional programming with python lambda expression we can solve it 

9 is the most sigintfite digite 
[9] 
Next 5 is the next singinfate digite 
[9,5] 
Next 30 ,34 and 3 share the same signfite gifie (3)  we move to the next one 
So it will be [9,5,34]
Next 3 and 30 
So it will be [9,5,34,3,30]


Q- what you opinion (as mentor ) to my thinking process here ?