A deck-based game is a game derived from a mathematical game by placing instances of its moves on a deck of cards. Initial work on deck-based games demonstrated that imposing the deck formalism can completely change the nature of the game. In this study, an evolutionary algorithm is used to design decks for a deck-based version of John Nash's classic game divide-the-dollar. The deck-based game is called FRAX and is used for helping students learn the arithmetic of fractions. The game includes a version that uses fraction multiplication, but focuses on addition since this is the more difficult of the basic operations with fractions. The deck design algorithm uses a novel evolutionary strategy called the horde of dumb agents technique to compare and evaluate different decks. Evolution, within the horde of dumb agents strategy, is a uniquely valuable tool for understanding how naive players might approach a new game. It is shown that the techniques presented in this study are able to obtain useful information about decks and new design principles for FRAX decks are discovered. These discoveries include an heuristic for deck difficulty based on strategically matching and non-matching subsets of the cards.
