Original Math Notes
All Seasons
Episode 316: Contenders--Explore the Math
Original Math Notes
Episode 316: Contenders
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 2:
INT. EPPES' HOUSE / LIVING ROOM - NIGHT
CHARLIE and AMITA are playing poker.  He's wearing his best
pajamas.  She's got a sexy Teddy under a silk robe. Charlie
lays down his cards.

                 CHARLIE
          Beat three sixes.
          
AMITA puts down her cards.

                 AMITA
          Flush.  All hearts.
          
                 CHARLIE
          But... you should have folded.
          
                 AMITA
          I was feeling lucky.
          
                 CHARLIE
          Lucky?  There was a less than 18
          percent probability you'd complete
          the flush.

Poker

This Demonstration generates random poker hands. Each of the 2598960 different possible poker hands will get one of the ten possible rankings, from royal flush to nothing.
Scene 4:
ALAN picks up the cards, shuffles absently.

ENTER CHARLIE VISION: The cards shuffle in slow motion.
Equations are parsed into the flipping cards. The deck is
cut. THE ACE OF DIAMONDS.

EXIT VISION.

                 CHARLIE
             (to Amita)
          If we can construe each murder as a
          separate Markov Chain.  What if we
          tried to shuffle them together?
          
                 AMITA
          Analyze precipitating events and
          potential motives... Find any
          commonalities.

Transition Matrices of Markov Chains

Suppose that if it is sunny today, there is a 60% chance it will be sunny tomorrow and that if it is not sunny today, there is a 20% chance it will be sunny tomorrow. If we assume today's sunniness depends only on yesterday's sunniness (and not on previous days), then this system is an example of a Markov chain, an important type of stochastic process.
Scene 18:
Charlie is in the living room with FIVE OR SIX GRADUATE
STUDENTS.  There's a CHALKBOARD from the garage.

                 CHARLIE
          I think with that we'll call it a
          night.  And thanks again for dealing
          with the last minute change in venue.

Alan enters through the front door.  Is a little surprised by
the parade of STUDENTS exiting.

                 CHARLIE
          My Tuesday evening graduate topology
          seminar.
          
                 ALAN
          In our living room.
          
                 CHARLIE
          We usually meet in my office, but I'm
          kinda locked out at the moment.

Topology of Costa's Minimal Surface

Costa's minimal surface is described as a torus with three points removed. How is this done? The points are literally removed from 3D-space and flung to infinity. Here is a Demonstration of specifically where on the torus the points are removed and how the torus turns itself inside-out when the resulting ends of the surface are flung to infinity.
Scene 34:
                 CHARLIE
          There's a classic problem in
          combinatorics.  The Travelling
          Salesman Problem...
          
ENTER VISION:
A VINTAGE 50s GUY WITH A VACUUM CLEANER in front of a
SUBURBAN HOME.

                 CHARLIE
          The Salesman must visit a specific
          set of cities...

A MAP OF THE US. A ROUGH OCTAGON OF CITIES ARE HIGHLIGHTED.

                 CHARLIE
          Each leg of the trip has a different
          cost...
          
THE CITIES ARE CONNECTED BY RAIL LINES.  A PRICE APPEARS FOR
EACH CONNECTION.
          
                 CHARLIE
          And he must figure out the most
          routes that minimizes the total
          expense.

Traveling Salesman Problem

The traveling salesman problem asks for the shortest route by which a salesman can visit a set of locations and return home. Drag the points to change the locations the salesman visits to see how the route changes.
 
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