"The Cham-Cham" opens with a United States Air Force plane being shot down during a radio broadcast of the instrumental "Dangerous Game" by popular musical group the Cass Carnaby Five.
International Rescue suspect sabotage, and Lady Penelope, Tin-Tin and Parker travel to the Swiss Alps to investigate the band's current tour venue, the mountain resort Paradise Peaks.
On Tracy Island, Alan notes that each attack has occurred while popular band the Cass Carnaby Five have been performing their hit instrumental "Dangerous Game" on live radio.
Meanwhile, Jeff assigns Lady Penelope, Tin-Tin and Parker to investigate Paradise Peaks, a mountain-top hotel in the Swiss Alps that is currently playing host to Cass Carnaby and his group.
[6] In the morning, Penelope and Tin-tin ski down the mountain to Olsen's chalet and film him operating a strange machine that is decoding musical sounds into text stating the time of the next missile shipment.
The shipment seems doomed until Penelope, in the guise of Wanda Lamour, appears on stage and sings a lyrical version, devised by Brains, containing a new set of coded instructions.
Scriptwriter Alan Pattillo created its show business plot and the exotic setting of Paradise Peaks in an attempt to emulate classic Hollywood musicals.
[3] For the scene in "The Cham-Cham" where Penelope glides across the Paradise Peaks ballroom while singing "Dangerous Game", Webb worked the puppet from the stage while fellow operator Christine Glanville controlled its wired top portion from an overhead gantry.
[2] Gerry Anderson believed that Penelope and Tin-Tin's trip to Olsen's lodge looked suitably realistic, despite APF never having shown puppets skiing prior to this episode.
[1] Anderson himself conceived the "ski thrusters" used by the characters to ascend the mountain during their journey back to Paradise Peaks, in part to remove the need for the puppets to walk.
[5] The shots of Penelope and Tin-Tin skiing to Olsen's chalet are accompanied by an incidental track called "Happy Flying" that was originally composed for the Supercar episode "Amazonian Adventure".
[12] Tom Fox of Starburst magazine rates the episode 4 out of 5, describing the plot as "tenuous" but believing this to be redeemed by the production design and the scenes of the cable car rescue.
He praises the "confidence" of "The Cham-Cham", calling it a "triumph" for art director Bob Bell and writing that although the story has "occasional moments of silliness", "everything about the production works perfectly."
[12][7] Stephen La Rivière, author of Filmed in Supermarionation: A History of the Future, praises the episode's technical standards, remarking that the skiing and dancing sequences "[fly] in the face of what puppets can and can't do.
"[15] He further argues that the humour has intergenerational appeal, stating that Parker's double entendres are counterbalanced by overt slapstick moments such as the character's "Mary Poppins"-style descent using Penelope's umbrella.
[16] Heather Phares of AllMusic considers "Dangerous Game" to be a highlight of the release, commenting that while the instrumental version "[reflects] the Sixties' ongoing fascination with exotica and Latin pop", its lyrical counterpart "could be a kissing cousin to seductive spy themes like 'Goldfinger'.