The potential space of game designs is astronomically large. This paper shows how game design theories can be translated into a simple, tangible card deck that can assist in the exploration of new game designs within a broader "design space."By translating elements of game design theory into a physical card deck, we enable users to randomly sample a design space in order to synthesize new game design variations for a new play platform ("Lumies"). In a series of iterative design and testing rounds with various user groups, the deck has been optimized to merge relevant game theory elements into a concise card deck with limited categories and clear descriptions. In a small, controlled experiment involving groups of design students, we compare the effects of brainstorming with the card deck or the "Directed Brainstorming"method. We show that the deck does not increase ideation speed but is preferred by participants. We further show that our target audience, children, were able to use the card deck to develop dozens of new game ideas. We conclude that design space cards are a promising way to help adults and children to generate new game ideas by making it easier to explore the game design space.