There are at least two types of barrels. The more popular barrel
with columns of colored marbles from dark to light. This barrel is
very easy to solve. The other has 5 columns of 4 colored marbles
each and 3 space filling black marbles (see below). The latter is
more difficult to solve due to a lower symmetry and since only a
simultaneous shift of marbles in 3 columns is possible.

My Solution of the Barrel

To solve the barrel it necessary to find move sequences (I called
them tricks) which interchange only a few marbles. Once you have
these tricks, the barrel can easily be solved by suitably
selecting and executing them. Indeed, a single trick is sufficient
to solve the barrel.