Louis has three kinds of balls on the playground, and the green balls—which bounce the highest—seem to symbolize that special thing kids always want, but somehow never get. You know the special thing we're talking about. Every kid has something they super want but just can't seem to get their hands on.
"The green balls were the best" (15.6), Sachar explains simply, and poor Deedee does everything she can to get one, including jumping down the stairs, cutting across the lawn, and masquerading as a dead rat. Why are the green balls so good? Let's consider the other options.
The red balls "didn't bounce as high" (15.8), and the only other ball—a yellow one—is a dud Louis is always trying to get rid of: "It didn't bounce, and it never went the way it was kicked" (15.11). The green ball is clearly the best—and when Deedee finally gets to bounce a prized green ball, "it went fifty feet straight up in the air" (15.44) before she catches it. Fifty feet.
The red, yellow, and green balls represent the things kids want—and don't want—and also how ridiculously hard it is for kids to get what they want sometimes. On the flip side of this, the green ball reminds us that with a little cleverness and by keeping our eyes on the prize, anything is possible. Even getting a green ball at recess.