  We introduce $Recursive~Jigsaw~Reconstruction$, a technique for analyzing
reconstructed particle interactions in the presence of kinematic and
combinatoric unknowns associated with unmeasured and indistinguishable
particles, respectively. By factorizing missing information according to decays
and rest frames of intermediate particles, an interchangeable and configurable
set of $Jigsaw~Rules$, algorithms for resolving these unknowns, are applied to
approximately reconstruct decays with arbitrarily many particles, in their
entirety.
  That the Recursive Jigsaw Reconstruction approach can be used to analyze
$any$ event topology of interest, with any number of ambiguities, is
demonstrated through a twelve different simulated LHC physics examples. These
include the production and decay of $W$, $Z$, Higgs bosons, and supersymmetric
particles including gluinos, stop quarks, charginos, and neutralinos.
