Can you find a general rule for the maximum number of Sets possible using
n cards? Can you establish a theoretical upper bound for the number of
Sets? Is it always achievable?

Consider creating a game of Set with a different deck. What would you
change? The number of attributes? The number of possible values for each
attribute? What would be a Set for this new deck? Investigate the original
and warm up questions of this project for your new version of the game.

What is the average number of Sets among the first twelve cards on the
table? Here are some suggestions as you think about this problem. Playing
the game can lead to a good estimate of the average. Alternatively, you
can obtain an estimate by writing a computer simulation that counts the
number of Sets for randomly chosen groups of twelve cards. The analytic
solution to this problem is unknown. If we do not know the maximum
number of Sets possible for 12 cards, then we do not know the probabilities
of those events and cannot compute the expected value. The solution
offered in the article Developing mathematical reasoning using attribute
games computes the number of combinations of 3 cards out of 12. This
value counts many three-card groups that have two cards in common and
which, therefore, cannot both be Sets. The computed value is therefore an
upper bound for the expected value that is not accurate. Try calculating
the expected value for smaller groups of cards to get a sense of the
difficulties that arise.