  We derive exact results for the critical Casimir force (CCF) within the
one-dimensional Ising model with periodic boundary conditions (PBC's) and
long-range equivalent-neighbor ferromagnetic interactions of strength
$J_{l}/N>0$ superimposed on the nearest-neighbor interactions of strength
$J_{s}$ which could be either ferromagnetic ($J_{s}>0$) or antiferromagnetic
($J_{s}<0$). In the infinite system limit the model, also known as the
Nagle-Kardar model, exhibits in the plane $(K_s=\beta J_s,K_l=\beta J_l)$ a
critical line $2 K_l=\exp{\left(-2 K_s\right)}, K_s>-\ln3/4$, which ends at a
tricritical point $(K_l=-\sqrt{3}/2, K_s=-\ln3/4)$. The critical Casimir
amplitudes are: $\Delta_{\rm Cas}^{\rm (cr)}=1/4$ at the critical line, and
$\Delta_{\rm Cas}^{\rm (tr)}=1/3$ at the tricritical point. Quite unexpectedly,
with the imposed PBC's the CCF exhibits very unusual behavior as a function of
temperature and magnetic field. It is repulsive near the critical line and
tricritical point, decaying rapidly with separation from those two singular
regimes fast away from them and becoming attractive, displaying in which the
maximum amplitude of the attraction exceeds the maximum amplitude of repulsion.
This represents a violation of the widely-accepted "boundary condition rule",
which holds that the CCF is attractive for equivalent BC's and repulsive for
conflicting BC's independently of the actual bulk universality class of the
phase transition under investigation.
