$(a)$ What happens if a bar magnet is cut into two pieces: $(i)$ transverse to its length, $(ii)$ along its length?

$(b)$ A magnetised needle in a uniform magnetic field experiences a torque but no net force. An iron nail near a bar magnet, however, experiences a force of attraction in addition to a torque. Why?

$(c)$ Must every magnetic configuration have a north pole and a south pole? What about the field due to a toroid?

$(d)$ Two identical looking iron bars $A$ and $B$ are given, one of which is definitely known to be magnetised. (We do not know which one.) How would one ascertain whether or not both are magnetised? If only one is magnetised, how does one ascertain which one? [Use nothing else but the bars $A$ and $B$.]

$(a)$ In either case, one gets two magnets, each with a north and south pole.

$(b)$ No force if the field is uniform. The iron nail experiences a nonuniform field due to the bar magnet. There is induced magnetic moment in the nail, therefore, it experiences both force and torque. The net force is attractive because the induced south pole (say) in the nail is closer to the north pole of magnet than induced north pole.

$(c)$ Not necessarily. True only if the source of the field has a net nonzero magnetic moment. This is not so for a toroid or even for a straight infinite conductor.

$(d)$ Try to bring different ends of the bars closer. A repulsive force in some situation establishes that both are magnetised. If it is always attractive, then one of them is not magnetised. In a bar magnet the intensity of the magnetic field is the strongest at the two ends (poles) and weakest at the central region. This fact may be used to determine whether $A$ or $B$ is the magnet. In this case, to see which one of the two bars is a magnet, pick up one, (say, $A$) and lower one of its ends; first on one of the ends of the other (say, $B$), and then on the middle of $B$. If you notice that in the middle of $B$, A experiences no force, then $B$ is magnetised. If you do not notice any change from the end to the middle of $B$, then $A$ is magnetised.