You can find rule 118 in Chapter Two. Basically, rule 118 states that you have to rely on subterfuge to get what you really want. Subterfuge is any sort of deceit or trickery used to achieve one's goals. Here's rule 118 as stated by Bud:
You Have To Give The Adults Something That They Think They Can Use To Hurt You By Taking It Away. That Way They Might Not Take Something Away That You Really Do Want Unless They Are Crazy Or Real Stupid. They Won’t Take Everything Because If They Did They Wouldn’t Have Anything Left To Hold Over Your Head To Hurt You With Later.
In the story, Bud is staying with a foster family. The Amos family consists of Mr. and Mrs. Amos and their son, Todd. Todd is a bully, but he's good at hiding the evidence of his crimes. Somehow, he always manages to pin the blame on Bud, who is considerably smaller and younger than he is.
After one especially violent physical altercation, Todd once again manages to make it look like Bud is the instigator of the fight. Mrs. Owen is so angry that she proclaims that Bud can no longer stay in their home. Upon hearing this, Bud pretends to be upset at the thought of being sent back to the orphanage. In reality, he'd rather be back at the Home than live with the likes of Todd. So, he puts rule 118 to use.
Bud apologizes profusely to the whole family; as Mrs. Owen appears to be the one in charge, he reserves his most obsequious (excessively groveling) apologies for her. As a stroke of genius, he begs not to be sent back to the Home and to be given another chance. His aim is to draw attention away from the one thing which really means a lot to him: the suitcase that contains paraphernalia (articles or things) of sentimental value to him. As Mrs. Amos already knows how attached he is to his suitcase and the items in them, Bud doesn't want to give any further indication of the extent of his attachment.
By focusing Mrs. Amos' attention on his supposed desire NOT to return to the orphanage, he allows her to think that she has some power over him. He deludes her into thinking that his first concern is being allowed to stay. In line with rule 118, this leads Mrs. Amos to hold on to his suitcase instead of throwing it away. She thinks that she can use the suitcase later as a way to manipulate Bud. As the story continues, we read that Bud soon runs away from the Amos home, and as promised by rule 118, he has planned it just right. Bud does find the suitcase after escaping from the shed he is locked in. He manages to play one last prank on Todd before leaving with his beloved suitcase.