  We classify all contact structures with contact surgery number one on the
Brieskorn sphere Sigma(2,3,11) with both orientations. We conclude that there
exist infinitely many non-isotopic contact structures on each of the above
manifolds which cannot be obtained by a single rational contact surgery from
the standard tight contact 3-sphere. We further prove similar results for some
lens spaces: We classify all contact structures with contact surgery number one
on lens spaces of the form L(4m+3,4). Along the way, we present an algorithm
and a formula for computing the Euler class of a contact structure from a
general rational contact surgery description and classify which rational
surgeries along Legendrian unknots are tight and which ones are overtwisted.
