The four corner Cantor set C is a planar self-similar set generated by the IFS {(x/4, y/4), ((x+3)/4,y/4), (x/4,(y+3)/4), ((x+3)/4,(y+3)/4)}. In this talk I will give a complete characterization on the set of ts in which the projection C_t:={x+ty: (x,y)\in C} is a self-similar set having an exact overlap; the distribution of these ts in the real line is also considered. This is joint work with Beibei Sun.