Pure Lovelock gravity, which has only one N-th order term in action and equation of motion with no sum over lower orders, offers an interesting possibility that gravitationally we could have lived in any member of the dimensional spectrum, d=3N+1. This is because gravitational dynamics for compact objects like black holes as well as cosmology is the same in all these dimensions. For instance, gravitationally it is impossible to decide whether our Universe is four dimensional with Einstein gravity or seven dimensional with pure Gauss-Bonnet gravity.