What is the attractive force between a magnet and a magnetic material?

The force between two magnets is quite complicated and depends on the strength and orientation of both magnets and the distance and direction of the magnets relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field B of the other. The B field of a small magnet is proportional its magnetic moment m and is itself quite complicated.

Mathematically, the force on a small magnet having a magnetic moment 'm' due to a magnetic field B of another magnet is:

F = ? (m . B ),
 
where the gradient ? is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. (The dot product m · B = mBcos(?), where m and B represent the magnitude of the m and B vectors and ? is the angle between them.)

The same can be applied for a magnet and a magnetic (ferromagnetic material). In this case, 'm' will be the magnetic moment induced in the ferromagnetic material due to the presence of magnetic field 'B' of the magnet in its vicinity.

However, this can be applied with perfect accuracy only for point (zero size) magnets and magnetic materials.

Hope this helped.
All the best!

The force between magnets is certainly not a function of the square of the distance. For larger distances, it is closer to the 4th power of the distance: that is, when the distance doubles, the force decreases by 2^4 = 16!.

At first consideration, it might appear that the force should decrease as an inverse function of the square of the distance, but you would be missing a 'trick'.

The 'trick' is that you never have only a north or only a south pole. They ALWAYS come in pairs, so at any distance from the magnet, both poles of each magnet are interacting with both poles of other magnet, and any force that might appear at very close distances  is largely canceled out at distances much larger than the distance between the two poles of each magnet.

If you have a magnet that is 1/2" between the north and south poles, then, at say 1" between magnets, you are already getting a VERY substantial 'canceling' of the magnetic forces, but even at 1/4" apart, the reduction is much more dramatic that the square rule would suggest.