Bill is never listed as a regular for this series, but early in his career, as he was beginning to make a name for himself on I've Got A Secret, he made at least six consecutive appearances as a panelist from August 6 to September 10, 1952, and at least one more appearance a few months later.  It's also possible his appearances on this show started before August.  He was probably only a replacement for Meredith Willson during that time, but the regularity of his appearances warrant at least a mention on this site.    

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Many, maybe even all of Bill's appearances on The Name's The Same survive, as does much of the entire series.  The series has been seen on GSN: The Network For Games as recently as 2005 as part of their Black & White Overnight block.  

FOR MORE INFORMATION 
The Name's The Same page at Tim's TV Showcase 
Theme music copyright information at David Shields' site for Classic TV Themes