  We define a new class of plane billiards - the `pensive billiard' - in which
the billiard ball travels along the boundary for some distance depending on the
incidence angle before reflecting, while preserving the billiard rule of
equality of the angles of incidence and reflection. This generalizes so called
`puck billiards' proposed by M.Bialy, as well as a `vortex billiard', i.e. the
motion of a point vortex dipole in 2D hydrodynamics on domains with boundary.
We prove the variational origin and invariance of a symplectic structure for
pensive billiards, as well as study their properties including conditions for a
twist map, the existence of periodic orbits, etc. We also demonstrate the
appearance of both the golden and silver ratios in the corresponding
hydrodynamical vortex setting. Finally, we introduce and describe basic
properties of pensive outer billiards.
