```                 DAVID
(growing concerned)
Are you saying we're here on a
hunch?

THE GUARDS AND THE DRIVER --

CHARLIE
Not a hunch: an induction.
```

# Proof by Induction »

An induction proof of a formula consists of three parts. a) Show the formula is true for n = 1. b) Assume the formula is true for n. c) Using b), show the formula is true for n + 1.
```                 DON
risk/reward ratios.

CHARLIE
These patterns are hard-wired in
the brain. And not just humans:
the behavior of monkeys, fish --
even bees choosing flowers -- can
be modeled using the same
mathematical ratios.
(then --)
Fifty mice forage in a field...

AUDIENCE VISION - a field at night. The center is bare; the
margins are a little bushy, and the edges are totally
overgrown.

CHARLIE (cont'd)
How will the mice distribute across
the area?
```

# Optimal Patch Exploitation »

This Demonstration is an investigation into the efficiency of an animal (or organism) feeding in a circular food patch (gray circle) while traveling in a concatenated string of unit-length forward "hops". In between each hop, the animal executes precise turns through specific angles of all sizes that are chosen randomly from the probability distribution pictured in the upper left-hand corner. Food is distributed evenly inside the food patch, and every particle of food along the trajectory (the black line starting from the center of the patch) is "gathered" by the forager. Note that the animal does not find food in any place where it has previously traveled, or when it is outside of the food patch.