Chapter 12, Problem 32E

Exercises 31 through 34 refer to a variation of the chaos game. In this game you start with a square ABCD with sides of length 27 as shown in Fig. 12-41 and a fair die that you will roll many times. When you roll a 1, choose vertex A; when you roll a 2, choose vertex B; when you roll a 3, choose vertex C; and when you roll a 4 choose vertex D. (When you roll a 5 or a 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points $P_1$, $P_2$, $P_3$ $\left|elip\right|$ inside or on the boundary of the square according to the following rules.

• Start. Roll the die. Mark the chosen vertex and call it $P_1$

• Step 1. Roll the die again. From $P_1$ move two-thirds of the way toward the new chosen vertex. Mark this point and call it $P_2$.

• Steps 2, 3, etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex.

Figure 12-41

Using graph paper, find the points $P_1$, $P_2$, $P_3$ and $P_4$ corresponding to

a. the sequence of rolls 2, 2, 4, 4.

b. the sequence of rolls 2, 3, 4, 1.

c. the sequence of rolls 1, 3, 4, 1.