Updated: Oct 12, 2021
Broadcast on 10 March 1953, the 18th episode of the third series explored 'The Mystery of the Cow on the Hill'. The Radio Times listing is here on page 24, complete with a photo of our heroes posing with "a real Goon", as doodled by Larry Stephens.
Unfortunately, without a recording of any kind, we may never know what that bovine mystery was, or how it was solved.
Orchestra: Plays theme from The X Files...
One thing we do know, however, thanks to the work of Roger Wilmut, is that this episode featured a sketch entitled 'The Siege of Khartoum'. This was reused in the fourth series in episode 18, 'The History of Communications', and includes a recurring gag about Arsenal losing to Sunderland.
The fourth series version uses a recent actual result from Arsenal's disappointing 1953-54 campaign, so I can only guess at what Spike Milligan and Larry Stephens used initially. The Gunners beat Sheffield Wednesday 4-1 the day before this episode was broadcast, so that result may have been too late to make it into the recording. Perhaps they were still banging on about Arsenal losing to Blackpool in the cup?