One of the Buckeye State's major mid-20th century discount stores had humble beginnings in the early 1950s. Rink's Bargain City, promoted in this 1953 logo, opened its first store in a vacant skating rink...hence the "Rinks" name. 
Graphic from Rink's Bargain City


Meanwhile, up in Toledo, a discount chain originally known as Bargain Barn was also on the grow. By the time of this 1961 ad, the enterprise was (also) known as Bargain City. The two independently owned and operated retailers were merged in 1967.
Drawing and graphic from Bargain City Stores


This trademark dates to 1971.
Graphic from Rink's Discount Department Stores


A vintage aerial view of the Middletown, Ohio Rink's. It included a Fisher-Fazio supermarket (which was located in the smaller structure on the right). The combo discount department store and grocery opened in 1966.
Photo from the George C. Crout Collection, courtesy of the Middletown Public Library


By 1974, the enterprise was being promoted as Rink's Bargain City.
Graphic from Rink's Discount Department Stores


Moving right along, we have a snapshot a 1960s Buckeye Mart. This store was located in Greater Columbus' BERWICK PLAZA. In latter years, the building would house the first Big Lots store.
Photo from Wikipedia / "Dirtyharry667"


This defunct retailer is difficult to categorize. Was it a Northeastern or Midwestern concern? The first store (promoted in this ad) opened in Greater Youngstown, in April 1958. At its peak, the chain operated in twelve states. Although the corporate headquarters was in Canton, Massachusetts, Hills would never open a store in New England.  
Advert from Hills Department Stores, Incorporated


Hills opened a store in Middletown, Ohio in May 1972. It covered 74,000 square feet.
Photo from the George C. Crout Collection, courtesy of the MidPointe Library System Digital Archives 


Over the course of  this chain's 21-year-trajectory, its official logo changed three times. At first, it was spelled simply Turnstile. In the early '60s, this morphed into Turn-Style. The final version, from the early '70s, had the name written as Turn*Style (with an asterisk instead of a hyphen). On the logo, the asterisk was portrayed as a spinning turnstile.
Graphic from Jewel Tea Company, Incorporated  


A rendering from the early 1970s portrays a new Indianapolis Turn*Style Family Center. It includes an attached Eisner Food Stores market. 
Drawing from Jewel Tea Company, Incorporated Annual Report 1970