You find yourself on a reality TV show that has you competing with other real people in totally artificial circumstances. In one scenario, you are given nine balls of clay. You are informed by the program's B-celebrity host that hidden inside one of those clay balls is a key that will unlock a refrigerator that houses a vast quantity of food. Since the producers "thought" the ratings would be higher if the contestants were deprived of nutrition, even the thought of brussel sprouts makes your mouth water. You are told that the eight balls that do not contain the key to your dietary dreams all weigh the same. The special ball with the key inside weighs slightly more, but not enough for you to feel the difference by holding the balls in your hand. One of the program's sponsors, Replace-Oh!, the company that manufactures one-time-use balance scales ( with the slogan "weigh aweigh then throw away!"), has agreed to provide some of its scales in exchange for a few shameless plugs throughout the program. Their scales will tell which side is heavier and then instantly self-destruct. You are only allow to break open one clay ball to see if you can find the refrigerator key. Your challenge is to determine the fewest disposable balance scales required to guarantee that you can identify the ball with the key. Justify your answer.