We explore models of intersecting brane worlds with induced gravity terms on codimension-one branes and on their intersection. Maximally symmetric solutions for the branes and the intersection are found. We find new self-accelerating solutions. In a 6D spacetime, the solutions realize the seesaw modification of gravity where the UV scale of the modification to 4D gravity is determined by 6D Planck scale given by M6∼10-3 eV and the IR scale of the modification is determined by M62/M4∼H0∼10-42 GeV, where H0 is the present-day Hubble scale. We find that it is increasingly difficult to construct phenomenologically viable models in higher-dimensional spacetime due to the necessity to have the lower value for the fundamental Planck scale to realize the late time acceleration. It is found that the system also admits self-tuning solutions where the tension at the intersection does not change the geometry of the intersection. The induced gravity terms can avoid the necessity to compactify the extra dimensions. Finally, we discuss the possibility to have ordinary matter at the intersection, without introducing any regularization, using the induced gravity terms.