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 2011^{th} lexicographic maximix arrangement for eleven carriages.

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These problems are part of
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