Update articles

articles/anagram-groups.md

@@ -1,5 +1,20 @@
 ## 1. Sorting
 
+### Intuition
+
+Anagrams become identical when their characters are sorted.  
+For example, `"eat"`, `"tea"`, and `"ate"` all become `"aet"` after sorting.  
+By using the sorted version of each string as a key, we can group all anagrams together.  
+Strings that share the same sorted form must be anagrams, so placing them in the same group is both natural and efficient.
+
+### Algorithm
+
+1. Create a hash map where each key is the sorted version of a string, and the value is a list of strings belonging to that anagram group.
+2. Iterate through each string in the input list:
+   - Sort the characters of the string to form a key.
+   - Append the original string to the list corresponding to this key.
+3. After processing all strings, return all values from the hash map, which represent the grouped anagrams.
+
 ::tabs-start
 
 ```python
@@ -153,6 +168,23 @@ class Solution {
 
 ## 2. Hash Table
 
+### Intuition
+
+Instead of sorting each string, we can represent every string by the frequency of its characters.  
+Since the problem uses lowercase English letters, a fixed-size array of length `26` can capture how many times each character appears.  
+Two strings are anagrams if and only if their frequency arrays are identical.  
+By using this frequency array (converted to a tuple so it can be a dictionary key), we can group all strings that share the same character counts.
+
+### Algorithm
+
+1. Create a hash map where each key is a `26`-length tuple representing character frequencies, and each value is a list of strings belonging to that anagram group.
+2. For each string in the input:
+   - Initialize a frequency array of size `26` with all zeros.
+   - For each character in the string, increment the count at the corresponding index.
+   - Convert the frequency array to a tuple and use it as the key.
+   - Append the string to the list associated with this key.
+3. After processing all strings, return all the lists stored in the hash map.
+
 ::tabs-start
 
 ```python

articles/duplicate-integer.md

@@ -1,5 +1,16 @@
 ## 1. Brute Force
 
+### Intuition
+
+We can check every pair of different elements in the array and return `true` if any pair has equal values.  
+This is the most intuitive approach because it directly compares all possible pairs, but it is also the least efficient since it examines every combination.
+
+### Algorithm
+
+1. Iterate through the array using two nested loops to check all possible pairs of distinct indices.
+2. If any pair of elements has the same value, return `true`.
+3. If all pairs are checked and no duplicates are found, return `false`.
+
 ::tabs-start
 
 ```python
@@ -131,6 +142,20 @@ class Solution {
 
 ## 2. Sorting
 
+### Intuition
+
+If we sort the array, then any duplicate values will appear next to each other.  
+Sorting groups identical elements together, so we can simply check adjacent positions to detect duplicates.  
+This reduces the problem to a single linear scan after sorting, making it easy to identify if any value repeats.
+
+### Algorithm
+
+1. Sort the array in non-decreasing order.
+2. Iterate through the array starting from index `1`.
+3. Compare the current element with the previous element.
+4. If both elements are equal, we have found a duplicate — return `True`.
+5. If the loop finishes without detecting equal neighbors, return `False`.
+
 ::tabs-start
 
 ```python
@@ -255,6 +280,22 @@ class Solution {
 
 ## 3. Hash Set
 
+### Intuition
+
+We can use a hash set to efficiently keep track of the values we have already encountered.  
+As we iterate through the array, we check whether the current value is already present in the set.  
+If it is, that means we've seen this value before, so a duplicate exists.  
+Using a hash set allows constant-time lookups, making this approach much more efficient than comparing every pair.
+
+### Algorithm
+
+1. Initialize an empty hash set to store seen values.
+2. Iterate through each number in the array.
+3. For each number:
+   - If it is already in the set, return `True` because a duplicate has been found.
+   - Otherwise, add it to the set.
+4. If the loop finishes without finding any duplicates, return `False`.
+
 ::tabs-start
 
 ```python
@@ -387,6 +428,20 @@ class Solution {
 
 ## 4. Hash Set Length
 
+### Intuition
+
+This approach uses the same idea as the previous hash set method: a set only stores unique values, so duplicates are automatically removed.  
+Instead of checking each element manually, we simply compare the length of the set to the length of the original array.  
+If duplicates exist, the set will contain fewer elements.  
+The logic is identical to the earlier approach — this version is just a shorter and more concise implementation of it.
+
+### Algorithm
+
+1. Convert the array into a hash set, which removes duplicates.
+2. Compare the size of the set with the size of the original array.
+3. If the set is smaller, return `True` because duplicates must have been removed.
+4. Otherwise, return `False`.
+
 ::tabs-start
 
 ```python

articles/is-anagram.md

@@ -1,5 +1,19 @@
 ## 1. Sorting
 
+### Intuition
+
+If two strings are anagrams, they must contain exactly the same characters with the same frequencies.  
+By sorting both strings, all characters will be arranged in a consistent order.  
+If the two sorted strings are identical, then every character and its count match, which means the strings are anagrams.  
+
+### Algorithm
+
+1. If the lengths of the two strings differ, return `False` immediately because they cannot be anagrams.
+2. Sort both strings.
+3. Compare the sorted versions of the strings:
+   - If they are equal, return `True`.
+   - Otherwise, return `False`.
+
 ::tabs-start
 
 ```python
@@ -133,6 +147,24 @@ class Solution {
 
 ## 2. Hash Map
 
+### Intuition
+
+If two strings are anagrams, they must use the same characters with the same frequencies.  
+Instead of sorting, we can count how many times each character appears in both strings.  
+By using two hash maps (or dictionaries), we track the frequency of every character in each string.  
+If both frequency maps match exactly, then the strings contain the same characters with same frequencies, meaning they are anagrams.
+
+### Algorithm
+
+1. If the two strings have different lengths, return `False` immediately.
+2. Create two hash maps to store character frequencies for each string.
+3. Iterate through both strings at the same time:
+   - Increase the character count for `s[i]` in the first map.
+   - Increase the character count for `t[i]` in the second map.
+4. After building both maps, compare them:
+   - If the maps are equal, return `True`.
+   - Otherwise, return `False`.
+
 ::tabs-start
 
 ```python
@@ -310,6 +342,24 @@ class Solution {
 
 ## 3. Hash Table (Using Array)
 
+### Intuition
+
+Since the problem guarantees lowercase English letters, we can use a fixed-size array of length `26` to count character frequencies instead of a hash map.  
+As we iterate through both strings simultaneously, we increment the count for each character in `s` and decrement the count for each character in `t`.  
+If the strings are anagrams, every increment will be matched by a corresponding decrement, and all values in the array will end at zero.  
+This approach is efficient because it avoids hashing and uses constant space.
+
+### Algorithm
+
+1. If the lengths of the strings differ, return `False` immediately.
+2. Create a frequency array `count` of size `26` initialized to zero.
+3. Iterate through both strings:
+   - Increment the count at the index corresponding to `s[i]`.
+   - Decrement the count at the index corresponding to `t[i]`.
+4. After processing both strings, scan through the `count` array:
+   - If any value is not zero, return `False` because the frequencies differ.
+5. If all values are zero, return `True` since the strings are anagrams.
+
 ::tabs-start
 
 ```python

articles/is-palindrome.md

@@ -1,5 +1,21 @@
 ## 1. Reverse String
 
+### Intuition
+
+To check if a string is a palindrome, we only care about letters and digits—everything else can be ignored.  
+We can build a cleaned version of the string that contains only alphanumeric characters, all converted to lowercase for consistency.  
+Once we have this cleaned string, the problem becomes very simple:  
+a string is a palindrome if it is exactly the same as its reverse.  
+
+### Algorithm
+
+1. Create an empty string `newStr`.
+2. Loop through each character `c` in the input string:
+   - If `c` is alphanumeric, convert it to lowercase and add it to `newStr`.
+3. Compare `newStr` with its reverse (`newStr[::-1]`):
+   - If they are equal, return `True`.
+   - Otherwise, return `False`.
+
 ::tabs-start
 
 ```python
@@ -152,6 +168,28 @@ class Solution {
 
 ## 2. Two Pointers
 
+### Intuition
+
+Instead of building a new string, we can check the palindrome directly in-place using two pointers.  
+One pointer starts at the beginning (`l`) and the other at the end (`r`).  
+We move both pointers inward, skipping any characters that are not letters or digits.  
+Whenever both pointers point to valid characters, we compare them in lowercase form.  
+If at any point they differ, the string is not a palindrome.  
+This method avoids extra space and keeps the logic simple and efficient.
+
+### Algorithm
+
+1. Initialize two pointers:
+   - `l` at the start of the string,
+   - `r` at the end of the string.
+2. While `l` is less than `r`:
+   - Move `l` forward until it points to an alphanumeric character.
+   - Move `r` backward until it points to an alphanumeric character.
+   - Compare the lowercase characters at `l` and `r`:
+     - If they don’t match, return `False`.
+   - Move both pointers inward: `l += 1`, `r -= 1`.
+3. If the loop finishes without mismatches, return `True`.
+
 ::tabs-start
 
 ```python

articles/longest-consecutive-sequence.md

@@ -1,5 +1,25 @@
 ## 1. Brute Force
 
+### Intuition
+
+A consecutive sequence grows by checking whether the next number (`num + 1`, `num + 2`, …) exists in the set.  
+The brute-force approach simply starts from every number in the list and tries to extend a consecutive streak as far as possible.  
+For each number, we repeatedly check if the next number exists, increasing the streak length until the sequence breaks.  
+Even though this method works, it does unnecessary repeated work because many sequences get recomputed multiple times.
+
+### Algorithm
+
+1. Convert the input list to a set for **O(1)** lookups.
+2. Initialize `res` to store the maximum streak length.
+3. For each number `num` in the original list:
+   - Start a new streak count at 0.
+   - Set `curr = num`.
+   - While `curr` exists in the set:
+     - Increase the streak count.
+     - Move to the next number (`curr += 1`).
+   - Update `res` with the longest streak found so far.
+4. Return `res` after checking all numbers.
+
 ::tabs-start
 
 ```python
@@ -178,6 +198,33 @@ class Solution {
 
 ## 2. Sorting
 
+### Intuition
+
+If we sort the numbers first, then all consecutive values will appear next to each other.  
+This makes it easy to walk through the sorted list and count how long each consecutive sequence is.  
+We simply move forward while the current number matches the expected next value in the sequence.  
+Duplicates don’t affect the result—they are just skipped—while gaps reset the streak count.  
+This approach is simpler and more organized than the brute force method because sorting places all potential sequences in order.
+
+### Algorithm
+
+1. If the input list is empty, return `0`.
+2. Sort the array in non-decreasing order.
+3. Initialize:
+   - `res` to track the longest streak,
+   - `curr` as the first number,
+   - `streak` as `0`,
+   - index `i = 0`.
+4. While `i` is within bounds:
+   - If `nums[i]` does not match `curr`, reset:
+     - `curr = nums[i]`
+     - `streak = 0`
+   - Skip over all duplicates of `curr` by advancing `i` while `nums[i] == curr`.
+   - Increase `streak` by `1` since we found the expected number.
+   - Increase `curr` by `1` to expect the next number in the sequence.
+   - Update `res` with the maximum streak found so far.
+5. Return `res` after scanning the entire list.
+
 ::tabs-start
 
 ```python
@@ -413,6 +460,27 @@ class Solution {
 
 ## 3. Hash Set
 
+### Intuition
+
+To avoid repeatedly recounting the same sequences, we only want to start counting when we find the **beginning** of a consecutive sequence.  
+A number is the start of a sequence if `num - 1` is **not** in the set.  
+This guarantees that each consecutive sequence is counted exactly once.
+
+Once we identify such a starting number, we simply keep checking if `num + 1`, `num + 2`, … exist in the set and extend the streak as far as possible.  
+This makes the solution efficient and clean because each number contributes to the sequence only one time.
+
+### Algorithm
+
+1. Convert the list into a set `numSet` for O(1) lookups.
+2. Initialize `longest` to track the length of the longest consecutive sequence.
+3. For each number `num` in `numSet`:
+   - Check if `num - 1` is **not** in the set:
+     - If true, `num` is the start of a sequence.
+     - Initialize `length = 1`.
+     - While `num + length` exists in the set, increase `length`.
+   - Update `longest` with the maximum length found.
+4. Return `longest` after scanning all numbers.
+
 ::tabs-start
 
 ```python
@@ -597,6 +665,33 @@ class Solution {
 
 ## 4. Hash Map
 
+### Intuition
+
+When we place a new number into the map, it may connect two existing sequences or extend one of them.  
+Instead of scanning forward or backward, we only look at the lengths stored at the **neighbors**:
+
+- `mp[num - 1]` gives the length of the sequence ending right before `num`
+- `mp[num + 1]` gives the length of the sequence starting right after `num`
+
+By adding these together and including the current number, we know the total length of the new merged sequence.  
+We then update the **left boundary** and **right boundary** of this sequence so the correct length can be retrieved later.  
+This keeps the whole operation very efficient and avoids repeated work.
+
+### Algorithm
+
+1. Create a hash map `mp` that stores sequence lengths at boundary positions.
+2. Initialize `res = 0` to store the longest sequence found.
+3. For each number `num` in the input:
+   - If `num` is already in `mp`, skip it.
+   - Compute the new sequence length:
+     - `length = mp[num - 1] + mp[num + 1] + 1`
+   - Store this length at `num`.
+   - Update the boundaries:
+     - Left boundary: `mp[num - mp[num - 1]] = length`
+     - Right boundary: `mp[num + mp[num + 1]] = length`
+   - Update `res` to keep track of the longest sequence.
+4. Return `res` after processing all numbers.
+
 ::tabs-start
 
 ```python