A method includes first revealing a limited set of potential game results to a player through a player terminal and then presenting the player with a set of graphic representations through the player terminal. Each respective representation has no apparent relationship to any particular potential result in the limited set of potential results. Thus, the player who is presented with the graphic representations cannot tell if any particular one of the representations is tied in any way to any particular potential result in the limited set of potential results. After presenting to the player the set of representations, the player is allowed to select at least one of the representations by making a player choice input through the player terminal. In response to the player choice input, at least one of the potential results in the limited set of potential results is assigned to the player, and this assigned potential result becomes the player's result for the game.