Surviving in a diverse environment requires corresponding organism responses. At the cellular level, such adjustment relies on the transcription factors (TFs) which must rapidly find their target sequences amidst a vast amount of non-relevant sequences on DNA molecules. Whether these transcription factors locate their target sites through a 1D or 3D pathway is still a matter of speculation. It has been suggested that the optimum search time is when the protein equally shares its search time between 1D and 3D diffusions. In this paper, we study the above problem using Monte Carlo simulations by considering a simple physical model. A 1D strip, representing a DNA, with a number of low affinity sites, corresponding to non-target sites, and high affinity sites, corresponding to target sites, is considered and later extended to a 2D strip. We study the 1D and 3D exploration pathways, and combinations thereof by considering three different types of molecules: a walker that randomly walks along the strip with no dissociation; a jumper that represents dissociation and then re-association of a TF with the strip at later time at a distant site; and a hopper that is similar to the jumper but it dissociates and then re-associates at a faster rate than the jumper. We analyze the final probability distribution of molecules for each case and find that TFs can locate their targets on the experimental time scale even if they spend only 15% of their search time diffusing freely in the solution. This agrees with recent experimental results obtained by Elf et al (2007 Science 316 1191) and is in contrast to previously reported theoretical predictions. Our results also agree with the experimental evidence for the role of chaperons and proteasomes in stabilizing and destabilizing TFs binding, respectively, during the regulation process. Therefore, the results of our manuscript can provide a refined theoretical framework for the process.