Novel methods, devices and systems are described for mapping a variety of Class III game outcomes to a common set of bingo patterns. Each game theme may have a different entertaining display, based upon a corresponding Class III game. Preferably, each game theme will offer game play and paytable percentages closely matching those of the original Class III game. Some implementations provide a system wherein electronic gaming machines presenting entertaining displays of various Class III game themes are linked to a single bingo server. By linking many participating electronic gaming machines to a single server, some implementations of the invention allow all of the progressive contributions to be pooled into one large progressive jackpot, thereby making the game more attractive to players.