Magnetic field strength attenuation: how to estimate if a magnet will affect another across a barrier such as wood or stainless steel?

I need to be able to get a compass needle (a magnet) to be affected (i.e., moved) by another magnet ("magnetically coupled"). These magnets are on opposing sides of a solid barrier of thickness $t$.

What properties of the magnets and of the barrier are needed to estimate if the magnets will be able to "feel" each other through the barrier?

What is the mathematical relationship between these parameters? That is, given a constant set of magnet parameters, can I estimate when the barrier is too thick for the magnets to affect each other?

How does the orientation of the magnets come into play? E.g. if the poles of the magnets are oriented normal to the barrier wall versus oriented parallel to the barrier wall?