Actually, Adam Nedeff and I (your two intrepid Cullen chroniclers on the WWW) were all over this, based on an earlier Cullen photo auction that Adam discovered.
The press release that must have come with the photo was touting a special celebrity week of shows, which aired October 17-21, 1966. It was a big week for NBC's daytime lineup, what with the debut of The Pat Boone Show as well as the premiere of some sort of new tic-tac-toe game played with celebrities.
Celebrities for the Eye Guess week were as follows:
Monday: Joan Fontaine and Darren McGavin
Tuesday: Betsy Palmer and Barry Nelson
Wednesday: Julia Meade and Mel Brooks
Thursday: Marty Allen and Steve Rossi
Friday: Bill and his wife Ann as contestants, Jack Clark hosting
Interestingly, Mel Brooks was such a low-wattage star at that point, he's not mentioned in the press release.
This'll all turn up on a way-overdue Bill Cullen Homepage update soon.