# Problem 336: Maximix Arrangements
A train is used to transport four carriages in the order: ABCD. However,
sometimes when the train arrives to collect the carriages they are not
in the correct order. To rearrange the carriages they are all shunted on
to a large rotating turntable. After the carriages are uncoupled at a
specific point the train moves off the turntable pulling the carriages
still attached with it. The remaining carriages are rotated 180 degrees.
All of the carriages are then rejoined and this process is repeated as
often as necessary in order to obtain the least number of uses of the
turntable. Some arrangements, such as ADCB, can be solved easily: the
carriages are separated between A and D, and after DCB are rotated the
correct order has been achieved. However, Simple Simon, the train
driver, is not known for his efficiency, so he always solves the problem
by initially getting carriage A in the correct place, then carriage B,
and so on. Using four carriages, the worst possible arrangements for
Simon, which we shall call maximix arrangements, are DACB and DBAC; each
requiring him five rotations (although, using the most efficient
approach, they could be solved using just three rotations). The process
he uses for DACB is shown below. It can be verified that there are 24
maximix arrangements for six carriages, of which the tenth lexicographic
maximix arrangement is DFAECB. Find the 2011th lexicographic maximix
arrangement for eleven carriages.