The present invention is a gaming device and preferably a bonus round game of a gaming device that enables a player to initiate a randomly generated number, wherein said number determines the number of positions that a position marker will move along an enclosed path. Each time the marker lands upon a previously unmarked position, the game provides an award associated with that position to the player. When the marker lands upon a previously marked position, the game ends. The present invention also contemplates updating or changing the values of unmarked or unselected positions along the enclosed path by preferably adding to each unmarked or unselected position the value of the award of a previously selected position. That is, when the game randomly selects an unmarked position, the game provides the player with an award, wherein the game preferably replaces a previously achieved award with the award of the newly selected position. In an alternative embodiment of the invention, an accumulator symbol may be generated by the move generator which enables the player to receive an offer associated with more than one display position.