**
**

The goal of the Chaos Game is to improve students' geometric intuition
and algorithmic thinking.
Before playing **The Chaos Game**, you should understand the role
of the chaos game in constructing fractals. See the book
Fractals,
especially Lessons 5-7, for more details.

** How to play the Chaos Game.**
When you open the chaos game applet, you see a game board that consists
of the Sierpinski triangle computed down to level 2, i.e., with nine
smaller triangles. One of these triangles is colored green; this is
the **Target**. You also see a point colored red at the lower
right hand corner of the triangle. This is the **Seed**. Your
goal is to move the **Seed** into the INTERIOR of the **Target**
in as few moves as possible. Each time you make a "move" in the chaos
game, this point will move to a new location in the game board. This
becomes your current location.

** The moves.**
There are three possible moves in this chaos game. If you click on
the top (red) vertex, your current location moves half the distance to
the topmost vertex. Simialrly, clicking on the lower left (blue) or
lower right (green) vertex, moves the current location half the
distance to that vertex. By a judicious choice of moves, you should
be able to move your point into the INTERIOR of the **Target** in just 4
moves. This is indicated in the **Best Score** window. Your score
(the number of moves you have made) is recorded in the window called
**Your Score**.

In the upper left corner of the screen your successive moves are
recorded as a series of red, green, and blue dots. When you succeed
in moving your current location into the interior of the **Target**, you
receive a message telling you so.

** Resetting the Target.** To replay the
current game again (without changing **Target**, click on **Try
Again**. To change to a new **Target**, click on **Restart**. The
computer randomly selects a new **Target**.

** The Algorithm.** There is an algorithm
for moving the starting point into the interior of the **Target** in
exactly 4 moves, no matter what **Target** you start with. Your job is to
discover this algorithm. You should be able to explain in advance how
to move the **Starting Point** into the **Target** in a couple
of sentences, no matter where the **Target** is located.

** More difficult Targets..** If you
click on the **Medium**, **Hard**, or **Master** buttons at
the top of the game board, you find chaos games with a **Target**
at deeper
levels of the Sierpinski triangle. The **Best Score** panel is
appropriately modified. The algorithm that you developed at the
**Novice** level should help you maneuver through these more
difficult games.

** Other Chaos Games.** The default chaos
game is chaos game #1. If you click on **2** or **3** at the
upper right corner, you can play other (more difficult) chaos games.
The rules of games 2 and 3 involve rotations. In game 2, the rules
are the same at the two lower vertices. That is, if you click on
either blue or green, you move half the distance to the appropriate
vertex. But if you click on red, you first move half the distance to
the little red "**x**" and then rotate about the **x** by 180
degrees. That is, you flip around this vertex to the exact opposite
side.

For game three, you move half the distance to the corresponding red
**x** and then rotate by 180 degrees around it for each of the
three different moves.

**Bug Report:** Apparently there is a bug in Game two. For a
very few targets, the computer claims that you won when, in fact, you
only hit the boundary of the target, not the interior. This will be
fixed some time.... Thanks to Paul Strickland for pointing this out!

Go to The Chaos Game.

Created by Noah D. Goodman, Adrian Vajiac, and Robert L. Devaney, based on an idea of Kevin Lee.

For comments and suggestions write to Robert L. Devaney at bob@bu.edu