Roy Reynolds was an intellectual adversary of Earth-One's Batman, as such he became known as the Getaway Genius plotting perfect crimes with the means of extracting his henchmen from the clutches of the Dynamic Duo and police.
The Puzzler was an unidentified criminal who was highly competitive. Instead of getting out of jams, he wanted to place his foes into precarious situations, leading to a repeated enmity with Earth-Two's Superman and Lois Lane.
While the Genius found his own lackey's were his undoing, as they captured Batman, for the Puzzler it was his unbridled overconfidence.
Although minor footnotes in the criminal annals now, these two masterminds were at the top of their game, until they clashed with the World's Finest heroes too many times.