  We resolve the computational complexity of two problems known as
NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are
PPA-complete. For NECKLACE SPLITTING, this result is specific to the important
special case in which two thieves share the necklace. We do this via a
PPA-completeness result for an approximate version of the CONSENSUS-HALVING
problem, strengthening our recent result that the problem is PPA-complete for
inverse-exponential precision. At the heart of our construction is a smooth
embedding of the high-dimensional M\"obius strip in the CONSENSUS-HALVING
problem. These results settle the status of PPA as a class that captures the
complexity of "natural" problems whose definitions do not incorporate a
circuit.
