$B_{u,d}$ ${\to}$ $\pi K$ and $K \bar{K}$ decays, including the $\pi K$ puzzle and pure annihilation decay $B_{d}$ $\to$ $K^- K^+$, are studied in detail. Combining the latest experimental data on the $CP$-averaged branching ratios, direct and mixing-induced $CP$-asymmetries, total 14 observables (see Table.\[pikbr\], \[pikdcp\], \[pikmcp\]) for seven $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$ decay modes \[see Eq.(\[bm2pimkz\]—\[b02kzkz\])\], the fit on four parameters ($\rho^f_A$, $\phi^f_A$) and ($\rho^i_A$, $\phi^i_A$) is performed with the fixed value $\lambda_B$ $=$ 0.2 GeV and the approximation ($\rho_H$, $\phi_H$) = ($\rho^i_A$, $\phi^i_A$), where ($\rho^f_A$, $\phi^f_A$), ($\rho^i_A$, $\phi^i_A$) and ($\rho_H$, $\phi_H$) are assumed to be universal for factorizable annihilation amplitudes, nonfactorizable annihilation amplitudes and spectator scattering corrections, respectively.

-   Scenario II: $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$ and $\pi \pi$ decays, including $\pi \pi$ puzzle, are studied. Combining the latest experimental data on the $CP$-averaged branching ratios, direct and mixing-induced $CP$-asymmetries, total 21 observables (see Table.\[pikbr\], \[pikdcp\], \[pikmcp\]) for ten $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decay modes \[see Eq.(\[bm2pimkz\]—\[amp7\])\], the fit on five parameters ($\rho^f_A$, $\phi^f_A$), ($\rho^i_A$, $\phi^i_A$) and $\lambda_B$ is performed with the approximation ($\rho_H$, $\phi_H$) = ($\rho^i_A$, $\phi^i_A$).

-   Scenario III: As a general scenario, to clarify the relative strength among ($\rho^f_A$, 