Update README.md

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@@ -72,6 +72,29 @@ def binary_search_recursive(array, item, higher=None, lower=0):
 
 <br>
 
+* or a slightly different version that does not carry `lower` and `higher`:
+
+<br>
+
+```python
+def binary_search_recursive(array, item):
+
+    start, end = 0, len(array)
+    mid = (end - start) // 2
+
+    while len(array) > 0:
+        if array[mid] == item:
+            return True
+        elif array[mid] > item:
+            return binary_search_recursive(array[mid + 1:], item)
+        else:
+            return binary_search_recursive(array[:mid], item)
+    
+    return False
+```
+
+<br>
+
 ---
 
 ### in a matrix
@@ -127,10 +150,8 @@ def sqrt(x) -> int:
             
             if num > x:
                 right = mid - 1
-                
             elif num < x:
                 left = mid + 1
-                
             else:
                 return mid
             
@@ -190,3 +211,63 @@ def peak_element(nums):
 
         return left
 ```
+
+<br>
+
+---
+
+### find a desired sum
+
+<br>
+
+* if the array is sorted:
+
+<br>
+
+```python
+def find_pairs_max_sum(array, desired_sum):
+
+    i, j = 0, len(array) - 1
+
+    while i < j:
+        this_sum = array[i] + array[j]
+        if this_sum == desired_sum:
+            return array[i], array[j]
+        elif this_sum > desired_sum:
+            j -= 1
+        else:
+            i += 1
+            
+    return False
+```
+
+<br>
+
+* if the array is not sorted, use a hash table:
+
+<br>
+
+```python
+def find_pairs_not_sorted(array, desired_sum):
+
+    lookup = {}
+
+    for item in array:
+        key = desired_sum - item
+
+        if key in lookup.keys():
+            lookup[key] += 1
+        else:
+            lookup[key] = 1
+
+    for item in array:
+        key = desired_sum - item
+
+        if item in lookup.keys():
+            if lookup[item] == 1: 
+                return (item, key)
+            else:
+                lookup[item] -= 1
+
+    return False
+```