  A generalized Kochen-Specker theorem is proved. It is shown that there exist
sets of $n$ projection operators, representing $n$ yes-no questions about a
quantum system, such that none of the $2^n$ possible answers is compatible with
sum rules imposed by quantum mechanics. Namely, if a subset of commuting
projection operators sums up to a matrix having only even or only odd
eigenvalues, the number of ``yes'' answers ought to be even or odd,
respectively. This requirement may lead to contradictions. An example is
provided, involving nine projection operators in a 4-dimensional space.
