  The $B\to\pi K$ decays with significant electroweak penguin contributions
show a puzzling pattern. We explore this "$B\to\pi K$ puzzle" through a
systematic strategy. The starting point, which is essentially unaffected by
electroweak penguins, is the determination of the angle $\gamma$ of the
unitarity triangle through the CP-violating $B^0_d\to\pi^+\pi^-$,
$B^0_d\to\pi^-K^+$ asymmetries, yielding $\gamma=(73.9^{+5.8}_{-6.5})^\circ$,
and the extraction of hadronic parameters through the measured $B\to\pi\pi$
branching ratios. Using arguments related to the SU(3) flavour symmetry, we
convert the hadronic $B\to\pi\pi$ parameters into their $B\to\pi K$
counterparts, allowing us to predict the $B\to\pi K$ observables in the
Standard Model. We find agreement with the data for those quantities that are
only marginally affected by electroweak penguins, while this is not the case
for the observables with sizeable electroweak penguin contributions. Since we
may also perform a couple of internal consistency checks of our working
assumptions, which are nicely satisfied for the current data, and find a small
sensitivity of our results to large non-factorizable SU(3)-breaking
corrections, the "$B\to\pi K$" puzzle may be due to new physics in the
electroweak penguin sector. We show that it can indeed be resolved through such
a kind of new physics with a large CP-violating phase. Further insights into
the electroweak penguins are provided by the $B^+\to\pi^0K^+$ and
$B_d^0\to\pi^0K_{\rm S}$ CP asymmetries, and in particular through correlations
with various rare $K$ and B decays.
