Summary: Proof: Let S be the number of complete (m \Gamma 1)simplexes in K m (colored with f0; : : : ; m \Gamma
1g), where the (m \Gamma 1)simplexes in each msimplex are considered separately, and counted as
+1 or \Gamma1, by their induced orientations. We argue that S = I and S = C.
To prove that S = I , consider the following cases. If an (m \Gamma 1)face is internal, then it
contributes 0 to S, since the contributions of the two msimplexes containing it cancel each
other. Obviously, an internal (m \Gamma 1)face contributes 0 to I . An external (m \Gamma 1)face in the
boundary of K m is counted the same, +1 or \Gamma1 by orientation, in both S and I . Therefore,
S = I .
To prove that C = S, consider an msimplex ø m , and look at the following cases. If ø m
contains two (m \Gamma 1)faces which are completely colored, then ø m is not completely colored and
contributes 0 to C. Note that ø m contributes 0 also to S, since the contributions of the two
faces cancel each other. If ø m contains exactly one (m \Gamma 1)face which is completely colored
(with f0; : : : ; m \Gamma 1g), then ø m must be completely colored and contributes +1 or \Gamma1, by
orientation, to C as well as to S. If ø m does not contain any (m \Gamma 1)face which is completely
colored, then ø m is not completely colored and therefore, it contributes 0 to C as well as to
S. Finally, note that ø m cannot contain more than two (m \Gamma 1)faces which are completely
We can now derive the oriented version of Sperner's Lemma.
Theorem C.1 (Oriented Sperner's Lemma) Consider an orientable divided image K m of
M(oe m ) under /, and a Sperner coloring Ø : K m ! M(oe m ). There exists an odd number of