Interactive Computations

Use the free Wolfram *Mathematica Player* to interact with the math
behind NUMB3RS.

CHARLIE Consider the game of Chicken, in which there are three Nash Equilibria.

A Nash equilibrium of a game is a
strategy combination such that no party can improve its situation by changing its
strategy, assuming the complementary strategies of the other players stay the same.

CHARLIE I had this idea that I could navigate the 11th Grade with the minmax theorem and n-person games. AMITA Did it work? CHARLIE My insights into the network externalities of school elections were bulletproof. Payoff Strategies for doing other people's homework were not quite as well reasoned.

In a strategic form game, each player's payoffs are a function of the combination of their probability distributions over their sets of strategies.

ALAN Charlie, didn't you do something like this before -- chase curves? CHARLIE Pursuit curves... and they're not really applicable --

Given a moving prey (black path) with unit speed and a time-dependent position {*x*(*t*), *y*(*t*)},
a predator (colored path), moving with speed *v*, runs at all times in the direction of the current position of the
prey.

CHARLIE ... a coastline at night. To keep us from crashing on the rocks, we build lighthouses. But lighthouses are a limited resource; they cost time, money, materials. Using Set Covering Deployment -- -- we determine the best placement of our limited number of lighthouses to illuminate the ocean.

Completely cover the orange shape
with the white disks. The number of disks determines the size of each shape.

ALAN Makes sense -- there must be some mathematical term for removing the...clutter--

Polynomials can be used to fit noisy data. This Demonstration allows you to explore
the relationship between the terms in a fit and the shape of the fit. You can experiment with different sets of powers.

CHARLIE Ernest Straus posited a roomful of mirrors... ... Straus wondered if there was a room so complex that a match lit in the right place couldn't reach every corner?

In the early 1950s, Ernst Straus asked if a single candle could illuminate an entire room made with
mirrored walls, no matter what the shape of the room.

CHARLIE It was forty years before George Tokarsky devised an answer -- a 26 sided room... (beat) ... and, in the spirit of that problem, and solution, I looked for Carter's dark corner -- the nearest, safest Chinese soil.

If a candle is inside a room with mirrored walls, can any portion of the room be dark? In 1958, a young Roger Penrose
found an unilluminable room with elliptical sides.