  We consider a swimmer consisting of a collinear assembly of three spheres
connected by two slender rods. This swimmer can propel itself forward by
varying the lengths of the rods in a way that is not invariant under time
reversal. Although any non-reciprocal strokes of the arms can lead to a net
displacement, the energetic efficiency of the swimmer is strongly dependent on
the details and sequences of these strokes, and also the sizes of the spheres.
We define the efficiency of the swimmer using Lighthill's criterion, i.e., the
power that is needed to pull the swimmer by an external force at a certain
speed, divided by the power needed for active swimming with the same average
speed. Here, we determine numerically the optimal stroke sequences and the
optimal size ratio of the spheres, while limiting the maximum extension of the
rods. Our calculation takes into account both far-field and near-field
hydrodynamic interactions. We show that, surprisingly, the three-sphere swimmer
with unequal spheres can be more efficient than the equally-sized case. We also
show that the variations of efficiency with size ratio is not monotonic and
there exists a specific size ratio at which the swimmer has the highest
efficiency. We find that the swimming efficiency initially rises by increasing
the maximum allowable extension of the rods, and then converges to a maximum
value. We calculate this upper limit analytically and report the highest value
of efficiency that the three-sphere swimmer can reach.
