  Given a Poisson point process on R, assign either one or two marks to each
point of this process, independently of the others. We study the motion of a
particle that jumps deterministically from its current location to the nearest
point of the Poisson point process which still contains at least one mark, and
removes one mark per each visit. A point of the Poisson point process which is
left with no marks is removed from the system. We prove that the presence of
any positive density of double marks leads to the eventual removal of every
Poissonian point.
