Pick a starting point. This point is the seed for the game. (Actually, the seed can be anywhere in the plane, even miles away from the triangle.) Then roll a die. Depending on the number that comes up, move the seed half the distance to the appropriately numbered vertex. That is, if 3 comes up, move the point half the distance to the vertex marked (3,4). Now erase the original point and begin again, using the result of the previous roll as the seed for the next. That is, roll the die again and move the new point half the distance to the appropriate vertex, and then erase the starting point.
Now continue in this fashion for a small number of rolls of the die. Five rolls are sufficient if you are playing the game ``by hand'' or on a graphing calculator, and eight are sufficient if you are playing on a high-resolution computer screen. (If you start with a point outside the triangle, you will need more of these initial rolls.)
After a few initial rolls of the die, begin to record the track of these traveling points after each roll of the die. The goal of the chaos game is to roll the die many hundreds of times and predict what the resulting pattern of points will be. Most students who are unfamiliar with the game guess that the resulting image will be a random smear of points. Others predict that the points will eventually fill the entire triangle. Both guesses are quite natural, given the random nature of the chaos game. But both guesses are completely wrong. The resulting image is anything but a random smear; with probability one, the points form what mathematicians call the Sierpinski triangle.
What do you think happens if you play with a square?