  The spread of entanglement is a problem of great interest. It is particularly
relevant to quantum state synthesis, where an initial direct-product state is
sought to be converted into a highly entangled target state. In devices based
on pairwise exchange interactions, such a process can be carried out and
optimized in various ways. As a benchmark problem, we consider the task of
spreading one excitation among $N$ two-level atoms or qubits. Starting from an
initial state where one qubit is excited, we seek a target state where all
qubits have the same excitation-amplitude -- a generalized-W state. This target
is to be reached by suitably chosen pairwise exchange interactions. For
example, we may have a a setup where any pair of qubits can be brought into
proximity for a controllable period of time. We describe three protocols that
accomplish this task, each with $N-1$ tightly-constrained steps. In the first,
one atom acts as a flying qubit that sequentially interacts with all others. In
the second, qubits interact pairwise in sequential order. In these two cases,
the required interaction times follow a pattern with an elegant geometric
interpretation. They correspond to angles within the spiral of Theodorus -- a
construction known for more than two millennia. The third protocol follows a
divide-and-conquer approach -- dividing equally between two qubits at each
step. For large $N$, the flying-qubit protocol yields a total interaction time
that scales as $\sqrt{N}$, while the sequential approach scales linearly with $
N$. For the divide-and-conquer approach, the time has a lower bound that scales
as $\log N$. With any such protocol, we show that the phase differences in the
final state cannot be independently controlled. For instance, a W-state (where
all phases are equal) cannot be generated by pairwise exchange.
