A two-dimensional numerical study of flow across rows of identical square cylinders arranged in staggered fashion is carried out. This study will unreveal complex flow physics depending upon the Reynolds number (Re) and gap spacing (g) between the cylinders. The combined effect of Reynolds number and gap spacing on the flow physics around staggered rows of cylinders are numerically studied for 20 ≤ Re ≤ 140 and 1 ≤ g ≤ 6. We use the lattice Boltzmann method for numerical computations. It is found that with increase in gap spacing between the cylinders the critical Reynolds number for the onset of vortex shedding also increases. We observed a strong effect of Reynolds number at g = 2 and 4. Secondary cylinder interaction frequency disappears for large Reynolds number at g = 6 and 5 and the flow around cylinders are fully dominated by the primary vortex shedding frequency. This ensures that at large gap spacing with an increase in the Reynolds number the wakes interaction between and behind the cylinders is weaken. Furthermore, it also ensures that the wake interaction behind the cylinders is strongly influenced by the jets in the gap spacing between the cylinders. We also found that g = 2 is the critical gap spacing for flow across rows of staggered square cylinders for the considered range of Reynolds number. Depending on the Reynolds number we observed; synchronous, quasi-periodic-I, quasi-periodic-II, and chaotic flow patterns. In synchronous flow pattern, an in-phase and anti-phase characteristics of consecutive cylinders has been observed. The important physical parameters are also analyzed and discussed in detail.