# Problem 597: Torpids
The Torpids are rowing races held annually in Oxford, following some
curious rules: A division consists of \$n\$ boats (typically 13), placed
in order based on past performance. All boats within a division start at
40 metre intervals along the river, in order with the highest-placed
boat starting furthest upstream. The boats all start rowing
simultaneously, upstream, trying to catch the boat in front while
avoiding being caught by boats behind. Each boat continues rowing until
either it reaches the finish line or it catches up with ("bumps") a boat
in front. The finish line is a distance \$L\$ metres (the course length,
in reality about 1800 metres) upstream from the starting position of the
lowest-placed boat. (Because of the staggered starting positions,
higher-placed boats row a slightly shorter course than lower-placed
boats.) When a "bump" occurs, the "bumping" boat takes no further part
in the race. The "bumped" boat must continue, however, and may even be
"bumped" again by boats that started two or more places behind it. After
the race, boats are assigned new places within the division, based on
the bumps that occurred. Specifically, for any boat \$A\$ that started
in a lower place than \$B\$, then \$A\$ will be placed higher than \$B\$
in the new order if and only if one of the following occurred: \$A\$
bumped \$B\$ directly \$A\$ bumped another boat that went on to bump
\$B\$ \$A\$ bumped another boat, that bumped yet another boat, that
bumped \$B\$ etc NOTE: For the purposes of this problem you may
disregard the boats' lengths, and assume that a bump occurs precisely
when the two boats draw level. (In reality, a bump is awarded as soon as
physical contact is made, which usually occurs when there is much less
than a full boat length's overlap.) Suppose that, in a particular race,
each boat \$B\_j\$ rows at a steady speed \$v\_j = -\$log\$X\_j\$ metres
per second, where the \$X\_j\$ are chosen randomly (with uniform
distribution) between 0 and 1, independently from one another. These
speeds are relative to the riverbank: you may disregard the flow of the
river. Let \$p(n,L)\$ be the probability that the new order is an even
permutation of the starting order, when there are \$n\$ boats in the
division and \$L\$ is the course length. For example, with \$n=3\$ and
\$L=160\$, labelling the boats as \$A\$,\$B\$,\$C\$ in starting order
with \$C\$ highest, the different possible outcomes of the race are as
follows: Bumps occurring New order Permutation Probability none \$A\$,
\$B\$, \$C\$ even \$4/15\$ \$B\$ bumps \$C\$ \$A\$, \$C\$, \$B\$ odd
\$8/45\$ \$A\$ bumps \$B\$ \$B\$, \$A\$, \$C\$ odd \$1/3\$     \$B\$
bumps \$C\$, then \$A\$ bumps \$C\$     \$C\$, \$A\$, \$B\$ even
\$4/27\$     \$A\$ bumps \$B\$, then \$B\$ bumps \$C\$     \$C\$, \$B\$,
\$A\$ odd \$2/27\$ Therefore, \$p(3,160) = 4/15 + 4/27 = 56/135\$. You
are also given that \$p(4,400)=0.5107843137\$, rounded to 10 digits
after the decimal point. Find \$p(13,1800)\$ rounded to 10 digits after
the decimal point.