With both attraction and repulsion abilities, the suspension of a magnet (by magnets) may appear at first a trivial matter: Strategically arrange a bed of magnets so that they repell a magnet placed on top of them.  Or perhaps position two layers of magnets such that both tug in equal magnitudes and opposite directions on a test magnet  placed between.  But do not pursue this futile venture long, for any attempt at the "special configuration" will always lead to failure. Says who?  Nature--or, if you will, her embassador James Clerk Maxwell, spoken in Nature's tongue as ·B = 0.   In man's tongue, this translates to: There can be no magnetic monopoles in Nature--that is, there does not and cannot exist local maxima of magnetic fields.  As far as levitating magnets goes, this means that a magnet cannot be suspended by other magnets because there will always be an area other than where the magnet is levitated that contains a lower potential energy (U = -m·B) to which the magnet is attracted.  So if a magnet is to be levitated, other forces of Nature must be appealed to in order to trap a magnet in space.

                Actually, the very forces which lead to the magnets instability can be exploited to achieve a stable magnet floating in space.  Because gravity provides a constant downward force, a magnetic field can provide a constant upward force on a magnet to result in a zero net force on the magnet, achieving verticle stability.  Since F =(m·B), a magnetic field which varies linearly will produce a constant, upwards force opposing gravity everywhere the gradient is constant (and of magnitude -(m/u)g ).  A pair of coils with radii a separated by a distance aSqrt(3) and possessing currents running in opposite directions will produce such a field.  These will be reffered to as the gradient coils. To ensure that the magnet is always aligned vertically--that is, anti-parellel with gravity--another pair of coils is necessary.  Helmoltz coils--coils of radii a separated by a distance a--will produce a uniform magnetic field within the region they bookend that will keep the magnet properly oriented wherever it finds itself between the coils.

                So now we have two pairs of coils which together will vertically orient the magnet and produce an upwards force to counter gravity.  But Maxwell's law has yet to be respected: The magnet cannot hover in free space with gravity out of the picture, but rather will seek out a lower potential energy than the state it is currently residing.  Where will it find this?  From the gradient and Helmholtz coils generating the fields the magnet is currently experiencing.  Hence, we have achieved vertical stability but have yet to conquer the lateral stability.  A magnet placed in the center of a set of gradient and Helmholtz coils appropriately stacked will quickly dart to the shelter of a coil's side where it will find a lower potential energy to inhabit.

                To trap the magnet in the center, the magnet must be convinced that the center is more stable than the wings of the coils.  Indeed, Maxwell's equation ·B=0 adamently prohibits any maximum magnetic field from being produced in the center of the coils.  But magicians can fool their audience into believing the most sacred, fundamental laws of Nature have been violated, yet of course not a single one ever is marred.  In the same spirit, a naive magnet can be duped into thinking that impossible has occured and a local maximum in the magnetic field has divinely arisen in the center of the coils.  The smoke and mirrors behind this trick is an oscillating quadupole potential produced by a pair of sinusoidally oscillating magnetic fields.  At one instant in time, one of these fields (again produced by coils) produces a field which contributes to the gradient--that is, vertically upwards--while the other, 180 out phase with the first, detracts from the gradient, encouraging the magnet to the side.  The former results in a stronger magnetic field in the center, and the off-centered magnetic is nudged back on target.  Maxwell's equation immedietely surges in protest, and indeed with good reason, for the magnet encounters instability as it returns to center.  And now for the slight of hand: Right before the magnet is able to zip off, the sinusoidally varying accomplices to the magnetic field flip so that the conditions are reversed.  Now, the direction which the magnet previously thought led to a lower potential is no more appealing while a luring deep potential appears as a mirage in middle.  Hence the magnet is literally pushed around by magnetic fields which continually decieve the magnet as to the location of a preferable potential.
 
 

The ploy can be pictured as a marble resting on a three dimensional saddle point.  Left to its own accord, the marble would happly roll off one of the sides into lower potential energy bliss.  But if the saddle rotates before the marble is able to roll anywhere, what was previously a luring downward slope turns into a daunting hill, while the appealing direction to roll is now in the opposite direction--which, inadvertantly, requires rolling towards the center.  Thus, by continually making the preferable location on the other side of the center, the marble will always want to roll towards the center in attempt to reach the ever changing, never approaching, lower potential.  The result: stability in the center of the saddle point.