Solve "find duplicate" using a graph

This problem is unusually difficult, but the solution is elegant.

scratch/facebook/find-duplicate-beast-mode.py

@@ -0,0 +1,57 @@
+def advance(position, xs):
+    """
+    Return the next element in `xs` pointed to by the current `position`.
+    """
+    return xs[position - 1]
+
+def find_duplicate(xs):
+    """
+    Find the duplicate integer in the list, `xs`.
+    """
+    beg = xs[-1]
+    a = beg
+    b = advance(a, xs)
+    # Find the first element of the cycle
+    cycle_beg = None
+    while a != b:
+        cycle_beg = a
+        a = advance(a, xs)
+        b = advance(b, xs)
+        b = advance(b, xs)
+    # The duplicate element is the element before the `cycle_beg`
+    a = beg
+    result = None
+    while a != cycle_beg:
+        result = a
+        a = advance(a, xs)
+    return result
+
+def find_duplicate(xs):
+    """
+    This is the solution that InterviewCake.com suggests.
+    """
+    # find length of the cycle
+    beg = xs[-1]
+    a = beg
+    for _ in range(len(xs)):
+        a = advance(a, xs)
+    element = a
+    a = advance(a, xs)
+    n = 1
+    while a != element:
+        a = advance(a, xs)
+        n += 1
+    # find the first element in the cycle
+    a, b = beg, beg
+    for _ in range(n):
+        b = advance(b, xs)
+    while a != b:
+        a = advance(a, xs)
+        b = advance(b, xs)
+    return a
+
+xs = [2, 3, 1, 3]
+result = find_duplicate(xs)
+print(result)
+assert result == 3
+print("Success!")