The boys tell Nanny that Arabella was insulted and then challenged to a race, but Nanny is having none of it. "Old cars are not meant to be raced." Professor Everett comes up with an alternative - feed the data into the computer. The problem is computers can't account for the human element.
Prudence's kite gets stuck in the tree of the local haunted house. The children discover how the house is being "haunted", by whom, and why. Then they have to decide whether to expose the information or keep the secret.
Prudence wishes for a new part in the school play on the first evening star. She finds that she doesn't like what happens for her wish to come true. It turns out to be the best outcome in the end and everyone is happy with their parts.
Professor Everett has to file a report on conservation of the Blackhawk Wilderness Area. Nanny discovers the children have never been on a campout. Each excuse the professor makes is mysteriously overcome and the family heads into the wilderness where nothing goes as planned.
Everett encounters a tough lady professor who espouses women's lib and leaves him frustrated. Meanwhile, a girl has keeps showing up the boys in their baseball matches, frustrating Butch and his friends, until Nanny attends a game and odd things conspire to make the girl unable to hit anymore, so she quits.
It's a rainy day and the children think it will never end. The professor tells them about a time when it rained two weeks and he and his brother built a canoe in the garage. The children decide to build a raft to sail on Squaw Lake.
Butch raises the ire of a neighbor as part of an initiation of the Oak Street Marauders. Peabody wanted to press charges, but the police talk him out of it. A visit by the whole family doesn't make progress, so Nanny comes up with an idea to try to warm Peabody up o the human race.
Nanny's globe-trotting Uncle Alfred turns up busking in the park, and is invited to stay with the Everetts. He tries to coax her to join him on a south sea voyage, and he grants supposedly magic wishes to the kids. They talk him into repeating former glories of when he did a human fly act for a local charity fair.
In previous years, Professor worked with an Italian director in Rome and is intimidated by his celebrity. The director is expected for a presentation at the university, but chooses to have dinner with the Everett family.
Nanny takes in some higher education by taking a course at Proffessor Everett's school, but Everett is wary of the teacher she gets, a hot shot playboy type who teaches the new wave psychology of relationships and sets his sights on making Nanny his next conquest.
Nanny's two oddball aunts come down in the Everett back garden in a hot air balloon, where they are determined to stay for a visit. The whole neighborhood gets involved, and a local Women's Lib group talks the pair into a passive protest by lying down in the street, leading to their arrest.