In this episode, a series of bank robberies have occurred, none of
them violent. After Charlie helps figure out where the next
robbery will occur, however, the robbers open fire and kill an
agent. By intervening, the FBI has changed the methods of the
robbers and revealed an even more dubious plot.

Charlie tells Don that the Heisenberg Uncertainty Principle says
that the act of measurement in a system affects the system. This
is, as Charlie admits later on, not actually what the principle
says. In fact, Charlie's statement is not especially profound:
when trying to measure very precisely the location of a piece of dust
floating in the air, we will naturally cause the air to move around and
thus cause the dust particle to move. This would not have been
news to physicists. What the Heisenberg Uncertainty Principle
says is that one cannot with perfect accuracy determine the position
and velocity of anything - especially not an electron. The
products of the errors is on the order of Planck's constant which is
about 10^{-34}Joules-seconds. A Joule-second is a unit
which is derived from only macroscopic quantities (like a kilogram and
a meter). So, in macroscopic terms, the error is completely
negligible. The smaller the things we're measuring, the more
important the principle becomes.

The Principle, at its most basic roots, has nothing to do with
physics at all. In fact, in its general form, it is a statement
about the relationship between a function and its Fourier transform.
It turns out that the Fourier transform is fundamental to quantum
mechanics, so naturally it has physical implications. The
American Mathematical Society has an excellent
article on the subject which also explains the Fourier transform as
well.

The P vs NP problem is a problem from the field of logic (or
theoretical computer science). Suppose we have a problem we want
a computer to solve. For example, we want a computer to find a
solution to the Travelling
Salesman problem. Suppose a salesman needs to travel to 50
cities across the globe. We know how much the flights cost
between each city and the salesman wants to spend as little as possible
flying. This clearly has a solution as there are "only"
50*49*48*....3*2*1 different paths to choose from. No one wants
to do this out by hand, so we want to program the computer to do
it. But how long will it take? This question is formulated
in the following way: suppose that we have n locations for the
problem. Is there an algorithm which requires O(f(n))
steps? Here the big-O notation means that for all sufficiently
large values of n, the number of steps required by our algorithm will
be smaller than our function f(n). In general, we say that a
problem is in an f-complexity class if there is an algorithm which
solves the problem in O(f(n)) steps.

The P-complexity class consists of all YES/NO problems which are
solvable in polynomial time. That is P consists of the set of
problems for which there exists an algorithm which can check a possible
solution to the problem in O(f(n)) for some polynomial f(n). We
say that P consists of problems for which there is a polynomial time
algorithm for checking solutions.
The NP-complexity class is somewhat more complicated. Basically,
though, the NP-class consists of problems for which there is a
polynomial time algorithm for finding
solutions. The P vs NP problem aks if P and NP are
actually equal. If the algorithm needed to verify a solution of a problem is in
polynomial time, is there an algorithm which solves the problem of finding a solution in polynomial
time? It seems like a somewhat silly question at first since
solving a problem and verifying a solution are almost the same thing to
a computer, but it is the one of the biggest open mathematical problems
in the world. The Clay
Mathematics Institute has offered $1,000,000 prize to anyone who
can give a positive or negative answer to whether P = NP.

The astute viewer noticed that Charlie says Minesweeper is
NP-complete. It is unclear what Charlie means by this since it
has very little bearing on solving the P vs NP problem. The
NP-complete problems represent a narrow subclass of NP problems which
might not be P. In essence, NP-complete are the hardest NP
problems. Richard Kaye proved Minesweeper is NP-complete in the
Mathematics Intelligencer. One can visit his webpage on the mathematics
of Minesweeper for more information. The Travelling Salesman
problem is also NP-complete.

At one point Charlie says to Don that he is statistically
dead. Having already been fired upon, Charlie implies that Don is
less likely to survive a second attempt. It is probably true that
a person who survives being fired upon once are more likely than the
average person to be killed in a gun fight during their lifetime.
After all, someone who is in a gunfight once is more likely to be in a
subsequent gun fight since this set of people includes police officers,
military personell, mercenaries, gang bangers, and so on. Is the
real implication Charlie is making correct? Specifically, is his
brother is more likely to die from a gunshot wound having already been
in a gunfight?