  We implement a variant of the quantum pigeonhole paradox thought experiment
to study whether classical counting principles survive in the quantum domain.
We observe strong measurements significantly violate the pigeonhole principle
(that among three pigeons in two holes, at least one pair must be in the same
hole) and the sum rule (that the number of pigeon pairs in the same hole is the
sum of the number of pairs across each of the holes) in an ensemble that is pre
and postselected into particular separable states. To investigate whether
measurement disturbance is a viable explanation for these counter-intuitive
phenomena, we employ the first ever variable-strength measurement of a
non-local observable. As we decrease the measurement strength, we find the
violation of the sum rule decreases, yet the pigeonhole principle remains
violated. In the weak limit, the sum rule is restored due to the cancellation
between two weak values with equal and opposite imaginary parts. We observe the
same kind of cancellation at higher measurement strengths, thus raising the
question: do strong measurements have imaginary parts?
