  A new family of free fermionic quantum spin chains with multispin
interactions was recently introduced. Here we show that it is possible to build
standard quantum Ising chains -- but with inhomogeneous couplings -- which have
the same spectra as the novel spin chains with multispin interactions. The
Ising models are obtained by associating an antisymmetric tridiagonal matrix to
the polynomials that characterize the quasienergies of the system via a
modified Euclidean algorithm. For the simplest non-trivial case, corresponding
to the Fendley model, the phase diagram of the inhomogeneous Ising model is
investigated numerically. It is characterized by gapped phases separated by
critical lines with order-disorder transitions depending on the parity of the
total number of energy density operators in the Hamiltonian.
