Why can't we make a perpetual motion machine by using a magnet to pull up a piece of metal, then letting it fall back down? Obviously, a perpetuum mobile isn't possible by any law in physics, because energy can't be "created" or "destroyed", only transformed.
This said, I've had an idea for a perpetuum mobile and can't seem to find my mistake (I'm actually very close to building and trying it).
Here's the plan:
Take a piece of wood and attach its upper end to a screw so it can swing like a pendulum. Then, attach a magnetic metal at the lower end of the wood. Now, place two magnets at each side of the pendulum, so that at the maximum amplitude, the metal will barely touch the magnets and will swing back, due to its weight.
Now, in my head, if you give the pendulum a little impulse, it will swing up in one direction and get attracted by the magnet just a tiny bit. Thus, on the "way back", it will have a slightly higher amplitude. So it swings to the other side, closer to the magnet, which will pull the pendulum a bit more upwards, thus increasing the amplitude furthermore.
This could theoretically go on and the pendulum will never stop, it will actually gain more momentum at the start.
So the conditions are:


*

*The Metal must be heavy enough so it doesn't stick to the magnets

*The Metal must be magnetic enough so that we gain amplitude instead of losing it each swing


And that's basically it. I am aware that the construction couldn't work, but struggle to find where I made my mistake.
Anyways, if it does work and you guys build it before I do: I want 50% of all profits and want you to name it Perpenduluum Mobile :D
 A: 
Now, in my head, if you give the pendulum a little impulse, it will swing up in one direction and get attracted by the magnet just a tiny bit.

You've neglected to account for the magnetic attraction as the pendulum bob goes back to its central position. 
On the outwards leg, you are correct that the magnet's attraction will pull on the bob and give it more energy than it would have in the absence of the magnet. However, in the return leg, the pendulum bob is trying to get away from the magnet's attractive force, and this will claim back all of the additional energy. 
(... if the system is perfect, that is. Real-world magnetic materials will show some amount of hysteresis, so the bob will lose slightly more energy on the way back than it gained on the way out.) 
This type of mistake is quite common when you have a core dynamics which is known to be conservative, and still seems to be producing energy - you're just conveniently neglecting to take into account the parts of the cycle where that force performs work against your system. For a similar example in action, see What prevents this magnetic perpetuum mobile from working?. 
A: Perpetual motion is impossible due to dissipation, or if you prefer second principle of thermodynamics  and not to energy conservation.
If your analysis of the suggested setup was correct, you could create mechanical energy for free !
In first analysis, neglecting the (unavoidable) losses in the ferromagnetic medium, your system is conservative : what you have is a modified pendulum, where the confinement potential contains not only the gravitational part but also a magnetic component. Actually the magnetic force slightly decrease the recall torque that you would have with gravity alone, and the amplitude of motion will indeed be larger. But you nevertheless will have a turning point where the kinetic energy vanishes, and when going back, you will reach exactly the same angle for the turning point on the other side.
This correspond to an oscillator with constant amplitude, because losses have been neglected.
The sources of losses are at least  : friction in air,  friction on axe, ferromagnetic hysteresis, Foucault currents. So the amplitude will decrease and the perpetual motion be reduced to perpetual immobility...
A: You are missing that as the metal passes through the changing magnetic field, Eddy currents will be produced.
These currents will result in the metal heating up (with the amount depending on the speed of travel and strength of field).
This heating essentially results in energy being removed from the system; and thus the bobbing can not be done forever; even if the rest of the system is perfect.
