The Partridge Family attend a lecture by Mike Rezo, a famous mystery writer. After the lecture they meet him and the discussion leads to the fact that everyone but Rezo feels that his stories are more fantasy than true to life. Rezo believes that using human nature and logic anything is possible. In order to prove his point, Rezo offers the Partridges a deal they can't refuse. He will give them a two-hour head start and then begin looking for them; if he can't locate them within 24 hours, he will donate $25,000 to their favorite charity.
The next morning the chase begins. No matter where they go, Rezo and his agent, Lazaar Hannibal, seem to turn up. The family pretends to leave their house and drive around for an hour before returning to the house – thinking that's the last place he'd look. When they get out of the car Rezo is waiting. They try getting lost in a crowd, hiding in an apartment of a friend of Reuben's, even the local ""lovers' lane,"" always to find Rezo waiting. Shirley leaves the younger children Chris and Tracy, at Mrs. Monihan's because the chase will be too much for them.
Trying to figure out how Rezo always finds them, Keith searches the car and finds a ""bug."" The Partridges use the bug to lead Rezo on a wild goose chase out to Cronos Caverns, 100 miles away. Thinking they are home free, the Partridges are surprised when Rezo walks in. He explains that the farther he got from town the weaker the signal from the bug became, and the closer he got to the Partridge home the stronger it became. Shirley says he still lost the bet since he hasn't found all the Partridges – where are Chris and Tracy?
At the show where Rezo turns the check for $25,000 over to the charity, Shirley learns from the younger children that Mr. Rezo had visited them at Mrs. Monihan's right after they arrived there. He had known all along, but donated the money anyway.