## Extension Problems for theGame of Set Problem

1. 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?

2. 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.

3. 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.

4. See the end of Developing mathematical reasoning using attribute games at http://www.setgame.com/set/article_nctm.htm for many more questions.

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