I'm trying to find a way to construct a bigger magnet from smaller ones. It seems like there is a way to be able to do it at least as far as I've read. I'm having trouble understanding how that would work if it is possible though. Say I have two square magnets and I want to make a rectangle magnet, if you put them side by side so that North is touching North and South is touching South. This would create a magnetic field with a split in the middle of the magnet wouldn't it? This would produce a different magnetic field than just making a new larger magnet, right?
Basically, you can add the separate magnetic fields of both magnets together. Far away from the magnets, that will look the same as a single large magnet, because the origins (the locations of the magnets) are so close together. Close to the magnets, things are indeed different.
If you model the magnet as a magnetic dipole, you could use a formula like this one and do some numerical calculations by adding two identical fields shifted by a distance $\Delta r$.
Whenever you bring two magnets in proximity applying force this means, that the energy you apply is stored in the magnetic field of the magnet: More field lines are pushed to the outside and the magnet becomes stronger. As the size of the magnet grows, the force density may grow to an smaller amount.
There is a good post about this from K&J Magnetics, Doubled Forces. This post shows that combining magnets can have a big effect on their pulling power but there is also a point where adding other magnets doesn't provide that much more useful force.