Abstract: Game systems, superficially similar to bingo, which employ combinations of multiple variables from different sets (for example colors; symbols such as numbers of letters; and geometric shapes such as circles, diamonds and squares) to increase the number of pattern permutations on individual game faces, in combination with random selection apparatus. Compared to ordinary bingo, the number of players may be greatly increased without resulting in games that are unduly short, and various version controls allow control over the mathematical probabilities of winning. Each game face has a plurality of individual positions organized as columns and rows. Within each column, the individual positions are filled with a random sequence of elements from each of at least two of the sets, and each column has a unique column designation. Random selection apparatus is disclosed, which is particularly useful when employed in combination with the game faces.