  We construct the exact solution for a family of one-half spin chains
explicitly. The spin chains Hamiltonian corresponds to an isotropic Heisenberg
Hamiltonian, with staggered exchange couplings that take only two different
values. We work out the exact solutions in the one-excitation subspace.
Regarding the problem of quantum state transfer, we use the solution and some
theorems concerning the approximation of irrational numbers, to show the
appearance of conclusive pretty good transmission for chains with particular
lengths. We present numerical evidence that pretty good transmission is
achieved by chains whose length is not a power of two. The set of spin chains
that shows pretty good transmission is a subset of the family with an exact
solution. Using perturbation theory, we thoroughly analyze the case when one of
the exchange coupling strengths is orders of magnitude larger than the other.
This strong coupling limit allows us to study, in a simple way, the appearance
of pretty good transmission. The use of analytical closed expressions for the
eigenvalues, eigenvectors, and transmission probabilities allows us to obtain
the precise asymptotic behavior of the time where the pretty good transmission
is observed. Moreover, we show that this time scales as a power law whose
exponent is an increasing function of the chain length. We also discuss the
crossover behavior obtained for the pretty good transmission time between the
regimes of strong coupling limit and the one observed when the exchange
couplings are of the same order of magnitude.
