This episode aired September 2, 1954 on the ABC Radio Network in the USA. 

http://www.enteringhemindseye.com/thefatman.html

The Fat Man, a popular radio show during the 1940s and early 1950s, was a detective drama created by Dashiell Hammett, author of The Thin Man. It starred J. Scott Smart in the title role, as a detective who started out anonymous but rapidly acquired the name 'Brad Runyon'. Broadcast from the studios of WJZ in Newark, New Jersey, the series premiered on the ABC Radio Network on Monday, January 21, 1946, at 8:30 p.m., as part of a block of four new programs and ran until 1951. J. Scott Smart (born John Kenley Tener Smart, November 27, 1902 – January 15, 1960) was an American radio, film and stage actor during the 1930s, 1940s and early 1950s. https://en.wikipedia.org/wiki/J._Scott_Smart

https://en.wikipedia.org/wiki/The_Fat_Man_(radio)