diff --git a/sudoku/__init__.py b/sudoku/__init__.py
index 130b175..3cb0572 100644
--- a/sudoku/__init__.py
+++ b/sudoku/__init__.py
@@ -68,8 +68,12 @@ class Sudoku:
         return self._possible_values
 
     def possible_values_for_square(self, x, y):
-        peers = self.peers(x, y)
-        return self.possible_values - peers
+        value = self.get(x, y)
+        if value:
+            return {value}
+        else:
+            peers = self.peers(x, y)
+            return self.possible_values - peers
 
     @property
     def rows(self):
diff --git a/sudoku/solvers/dlx.py b/sudoku/solvers/dlx.py
index e3657e0..39e4b98 100644
--- a/sudoku/solvers/dlx.py
+++ b/sudoku/solvers/dlx.py
@@ -124,7 +124,7 @@ def _build_headers(sudoku):
 def _build_rows(sudoku, headers):
     for (index, coords) in enumerate(sudoku.all_squares):
         board_value = sudoku.get(*coords)
-        possibilities = {board_value} if board_value else sudoku.possible_values_for_square(*coords)
+        possibilities = sudoku.possible_values_for_square(*coords)
         for value in possibilities:
             cur = None
             for col in headers.iterate_row():