  The Standard-Model interpretation of the ratios of charged and neutral B-> pi
K rates, R_c and R_n, respectively, points towards a puzzling picture. Since
these observables are affected significantly by colour-allowed electroweak (EW)
penguins, this ``B -> pi K puzzle'' could be a manifestation of new physics in
the EW penguin sector. Performing the analysis in the R_n-R_c plane, which is
very suitable for monitoring various effects, we demonstrate that we may, in
fact, move straightforwardly to the experimental region in this plane through
an enhancement of the relevant EW penguin parameter q. We derive analytical
bounds for q in terms of a quantity L, that measures the violation of the
Lipkin sum rule, and point out that strong phases around 90 deg are favoured by
the data, in contrast to QCD factorisation. The B -> pi K modes imply a
correlation between q and the angle gamma that in the limit of negligible
rescattering effects and colour suppressed EW penguins depends only on the
value of L. Concentrating on a minimal flavour-violating new-physics scenario
with enhanced Z^0 penguins, we find that the current experimental values on
  B -> X_s mu^+ mu^- require roughly L <= 1.8. As the B -> pi K data give L =
5.7 +- 2.4, L has either to move to smaller values once the B -> pi K data
improve or new sources of flavour and CP violation are needed. In turn, the
enhanced values of L seen in the B -> pi K data could be accompanied by
enhanced branching ratios for rare decays. Most interesting turns out to be the
correlation between the B -> pi K modes and BR(K^+ -> pi^+ nu nu), with the
latter depending approximately on a single ``scaling'' variable \bar L= L
(|V_{ub}/V_{cb}|/0.086)^2.3.
