In this episode Charlie uses graph theory to determine how power station failures will cause blackouts in Los Angeles.

There is a brief shot in this episode where the book "A New Kind
of Science" by Stephan Wolfram is positioned on Larry's desk in a way
that makes it look like he was recently reading it. This 1192 page
book was published in 2002 with mixed reviews at best. It was ciriticezed
by many respected scientists for many reasons. They said many of the author's
claimed results were not new and that he didn't provide any references to
previous related work done by many talented people. He showed poor
understanding of some areas of science, making some statements that were
completely wrong. The book also wasn't peer-reviewed, which is an important
part of publishing scientific works, and was published by the author's own
company. Since this episode was aired in 2006, it seems unlikely that
Larry would still be actively reading it, especially given its poor reviews
by respected scientists.
It is interesting to note that one of the employees of Wolfram's
company (which is responsible for Mathematica) is a mathematical consultant for
the Numb3rs show.

The first thing that Charlie needs to do is to describe the power station
and power transmission situations mathematically. A perfect tool for this
is a flow network, which can also be called a directed weighted graph.
We will represent power stations as squares, power users as circles, and
the power cables that transmit power as lines between circles and squares.
Each of these elements will also be assigned a number. Power stations
have a maximum amount of power that they can generate, and we label them
with this number. Power cables are labeled by their maximum transmission
capacities, and power users are labeled by the amount of power they want
to consume. The goal of the power companies is to choose how much electricity
from each power plant goes into each of the power lines that leave it
so that each power use gets as much power as they want.
Here is a simple example of such a graph. The black numbers in the boxes indicate the total generating capability of those power plants and the red numbers indicate how much electricity is actually sent along the corresponding power lines. You might notice that this system is pretty rigid, and if any power is knocked out, then at least one of the users will be without enough power.

Let's figure out exactly what will happen if one of the power plants fails. If the bottom power station stops working, then both the left and bottom users will be without electricity. If the middle power station stops working, the result will be the same. If the top left power station stops working, then the power operators will have a choice: they can either let the left user get 4 out of 5 units, or they can divert 1 unit from the right power station to the left user, but then the right user would get 4 out of 5 units. If the right station stops working, then the right user will get no electricity. The results are summarized in the table below.

Power Plant knocked out | Left User's Supply | Middle Users Supply | Right Users Supply |

Lower Left | 3/5 | 2/4 | 5/5 |

Middle | 3/5 | 2/4 | 5/5 |

Upper Left | 4/5 or 5/5 | 4/4 | 5/5 or 4/5 |

Upper Right | 5/5 | 4/4 | 0/5 |

In this episode Larry mentions a well known lecture given by the famous
physicist Richard Feynman in 1959 called "There's Plenty of Room at the
Bottom." In this lecture, Feynman predicted that there would be major
advances in nanotechnology and computing power in the next decades, and he
was completely right. He predicted the ability to write the entire
Encyclopedia Brittanica on the head of a pin, the ability to build miniscule
motors, the drastic miniaturization of computers, and several other major
advances. He also talked about several problems and differences of working
at such small scales. For his first two predictions he offered $1000 to the
first person who was able to write text on the head of a pin at 1/25,000 of
the size of normal text, and $1000 to the first person who made an operating
motor in a 1/64 inch cube. These prizes were claimed in the early 1960's and
in 1985, respectively. The content of this talk was just one more demonstration
of his genius. A transcript of the talk can be found
here.

This demonstrates one
inaccuracy in the show. Charlie claims that the criminal was able to
use complicated mathematics to predict exactly which areas would suffer
blackouts as a result of certain stations being knocked out. However, even
in this simple example there are choices the power operators would have to
make if the upper left station fails,
and real life power grids will be far more complicated than this graph.
This means it would be almost impossible to predict what choices would be
made unless one had inside knowledge about the protocols used to make these
decisions. Also, these protocols are very likely to use randomness, which
makes them harder to predict. For example, if the upper left power station
went out, the person in charge of distributing power might flip a coin
to decide whether to divert power from the right station or not, and then
switch every
day until the power came on to be more fair to the users.
Now we can use this example to demonstrate the logic that Charlie used in the show. Suppose that criminals knocked out the bottom and middle power plants on consecutive days. Then using Charlie's first idea and the assumption that every time they are knocking out a power plant they are trying to disrupt power for one specific user, we know that they must be trying to knock out power for either the left user or the bottom user. However, just like Charlie, we can't figure out which one it is exactly unless we use Larry's suggestion. He pointed out that if the criminals are trying to knock out power for one specific user, then the power stations they didn't disable can also tell us information about which user they are trying to affect. In this case, if the upper left station is broken, then the left user may or may not be affected, but the bottom user definitely will not. This means that since the criminals didn't disable the top left station, they must want to stop power from reaching the bottom users (since we can assume that they want to be certain that they affect the desired user.