  We consider the computational complexity of reconfiguration problems, in
which one is given two combinatorial configurations satisfying some
constraints, and is asked to transform one into the other using elementary
transformations, while satisfying the constraints at all times. Such problems
appear naturally in many contexts, such as model checking, motion planning,
enumeration and sampling, and recreational mathematics. We provide hardness
results for problems in this family, in which the constraints and operations
are particularly simple. More precisely, we prove the PSPACE-completeness of
the following decision problems:
  $\bullet$ Given two satisfying assignments to a planar monotone instance of
Not-All-Equal 3-SAT, can one assignment be transformed into the other by single
variable `flips' (assignment changes), preserving satisfiability at every step?
  $\bullet$ Given two subsets of a set S of integers with the same sum, can one
subset be transformed into the other by adding or removing at most three
elements of S at a time, such that the intermediate subsets also have the same
sum?
  $\bullet$ Given two points in $\{0,1\}^n$ contained in a polytope P specified
by a constant number of linear inequalities, is there a path in the n-hypercube
connecting the two points and contained in P?
  These problems can be interpreted as reconfiguration analogues of standard
problems in NP. Interestingly, the instances of the NP problems that appear as
input to the reconfiguration problems in our reductions can be shown to lie in
P. In particular, the elements of S and the coefficients of the inequalities
defining P can be restricted to have logarithmic bit-length.
