While searching for something, I have come across the fact that the gravitational acceleration in a mine shaft is independent of depth if the local density is $2/3$ of the average density (of the Earth).
Also, the gravitational acceleration increases only if the local density is greater than two-thirds of the average density. Note that everywhere a spherically symmetric model for the Earth was used.
Where does this figure come from?
This Question in Mentioned in Resnick Halliday Krane book as P-14-10.The reference is "Gravity in a Mine Shaft" by Peter M.Hall and David J.Hall,The Physics Teacer,November 1995 p.525.)