Question: Having a brick lying on a table, I can exert horizontal force equal to $\mu m g$ to a middle of it's side, and it will start moving (assume $\mu$ is the friction coefficient). However, can I make the brick moving with less horizontal force? May be, applying it not to a middle of a side can help? I have no idea of how to calculate or estimate it, but it would be interesting to know.

Answer: 
Suppose that you exert the force with angle $\theta$ (with respect to ground). Then you will have:

$$\mu(mg-F\sin(\theta))=F\cos(\theta)\text{, so }F=\frac{{\mu}mg}{\cos(\theta)+{\mu}\sin(\theta)}.$$

Now, if you minimize this function with respect to $\theta$ you will find that
$$\tan(\theta)=\mu.$$

Replacing this $\theta$ (a function of $\mu$) for $\sin(\theta)$ and $\cos(\theta)$ in the second formula (for $F$) you will have:
$$F\_{min}=\frac{\mu}{\sqrt{1+\mu^2}}mg.$$
If you can't change the brick or the surface(or the contact eg by lifting the brick)  

Then how you put the force into the brick doesn't change the energy treansferred