fix the DFS algorithm

Fix

Update:

- Implemented Depth First Search (DFS) to explore possible digit assignments for each letter, with pruning to discard any paths that can't yield a valid solution. This optimization speeds up runtime by eliminating unfeasible assignments early.
- Added backtracking to revert and try alternative assignments if a partial solution doesn’t lead to a valid final solution.

How This Works:
- The solver first extracts all unique letters from the puzzle.
- DFS explores each possible digit for these letters.
- Pruning: The solver skips digits already in use and only continues exploring if the partial assignment can still lead to a valid solution.
- Backtracking: If a particular assignment fails, the solver backtracks to try the next possible digit, ensuring all potential solutions are explored efficiently.

app.py

@@ -1,5 +1,4 @@
 import gradio as gr
-from itertools import permutations
 
 def solve_cryptarithm(equation):
     # Parse the words from the equation "SEND + MORE = MONEY"
@@ -28,8 +27,8 @@ def solve_cryptarithm(equation):
         
         # Try every digit for this letter
         for digit in range(10):
-            # Prune: skip used digits
-            if digit in used_digits:
+            # Prune: skip used digits and avoid leading zero for multi-letter words
+            if digit in used_digits or (digit == 0 and letter_is_leading[letter]):
                 continue
             # Map the letter to the current digit
             letter_to_digit[letter] = digit
@@ -44,8 +43,9 @@ def solve_cryptarithm(equation):
             used_digits.remove(digit)
         
         return False
-    
-    # Main DFS invocation
+
+    # Identify which letters are leading letters in each word
+    letter_is_leading = {word[0]: True for word in words}
     unique_letters_list = list(unique_letters)
     letter_to_digit = {}
     used_digits = set()
@@ -57,7 +57,7 @@ def solve_cryptarithm(equation):
         
         # Convert each word to its number representation
         for i, word in enumerate(words):
-            formatted_result += " ".join(str(letter_to_digit[char]) for char in word)
+            formatted_result += "".join(str(letter_to_digit[char]) for char in word)
             if i < len(words) - 2:
                 formatted_result += " + "
             elif i == len(words) - 2: