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# Probability of having a "perfect" game of Set

The card game Set has very simple rules (see here for rules), and it has prompted mathematicians to ask several questions. I will describe one of these questions. When the game ends, there are usually some left over cards, none of which form a Set; but occassionally it happens that the remaining cards all match up, and there are no left over cards. Call such a situation a "perfect" game. My question is:

What is the probability of having a perfect game? What are the best known upper and lower bounds for this probability?

Here we assume that if there are multiple sets available, then one is selected at random (with uniform probability among all distinct available sets). According to this handout the exact value for the probability of a perfect game is "very open". Anecdotally, I've probably played Set hundreds (maybe thousands?) of times, and only a handful have been perfect games (one of which was earlier today!).

A related question one could ask is whether or not the actual gameplay changes the probability. What I mean is:

Say all 81 cards are laid down face up, and Sets are removed randomly until no Sets remain. Is the probability of having no left over cards the same as it is when the game is played normally?

I could remark that the actual game of Set is a special case of a larger class of games, where the cards have n attributes each with r possible values; the usual game is the case n=4 and r=3. One could ask the same questions for this larger class of games. Also, these questions are more naturally viewed as questions about configurations of lines in affine space, but I've chosen to stick with the card game terminology.