Billy and Olie play a game where they become "invisible", trying to make no-one see them. For how long? They intend to go round the whole house being invisible!
The Bevilles arrive back to Polyville from their holiday to the square planet, Billy shows Olie and Polly his new "flying box". Olie and Polly are confused, when Billy is hiding something when the Bevilles go back into their house and Polly suspects they are moving...
While Olie, Billy and Uncle Gizmo are staying with Pappy out camping, Pappy tells them the story of Roundbeard, burying treasure nearby and how his ghost is still searching for it. Then, Olie, Billy and Gizmo think they see Roundbeard's ghost.
Olie loves Billy's new boxy toy plane and Billy kindly lets Olie borrow it while he's out. Olie plays around with the toy plane, but soon he breaks it and wonders if it can be fixed.
Olie and Billy are busy playing their games and momentarily forget that Zowie knows fun games too. But when they remember, they make up for it in spades as the Polies' backyard rocks to the merry sounds of dance music.
A planet made completely out of ice cream has been discovered; Uncle Giz's pomp is a little down, and so is he; Mom and Dad remember when they first met.
Billy feels like a square peg in a round hole. And Olie, being the best friend that he is, devises a square world for them to play in, much to Billy's delight.