## Hints for theGame of Set Problem

1. If you pull out 3 cards from the whole deck, what is the probability that they form a Set?
Try the Warm Ups and you should be ready for this question.

2. If we pull 9 cards from the deck, what is the maximum number of Sets possible among them?
The solution to this problem requires two parts. If you claim that the maximum number of Sets is n, then you need a proof showing that no more than n is possible and an example showing that at least n Sets can be made (or a proof showing that such a collection must be constructible even if you can’t find it).

3. How many cards can be left on the table at the end of the game?
This question is not the same as the question “How many cards can there be in a Set-less pile?” How does the way the game is played make it a different question? Consider just one attribute, for example, shading. At the beginning, there are 27 cards with each of the three possible shadings. What happens to the number of cards with each shading after a Set is taken away?

4. What can be found out about the maximum Set-less pile of cards?
Do not be frustrated if you cannot determine an exact answer for this question. A reasonable approach to this problem is to try to establish limits on the possible answer. What would you need to do to show that the maximum Set-less pile is at least some number n? What would you need to do to show that it is no bigger than some upper limit? If you can narrow down the possibilities, that is a worthy accomplishment.

One general approach is to start with a simpler problem: a smaller deck where cards have two or three attributes instead of four (or fewer values per attribute). Find the maximum Set-less pile in those cases and try to extend your methods as you add attributes.

Another approach: Imagine dividing the whole deck of Set into two piles: the first one, which is Set-less, and the second one, which contains all of the rest of the cards. Any two cards of the first pile make a Set with exactly one card of the deck. For the first pile to be Set-less, this card must be in the second pile. As you try to construct Set-less piles, it is helpful to think about the number of cards being forced into the complementary pile. How can you choose cards for your Set-less pile such that the second pile does not grow too large?

Alternatively, you could try to think geometrically about the problem. How can the different attributes and values be represented visually?

 Translations of mathematical formulas for web display were created by tex4ht. © Copyright 2003 Education Development Center, Inc. (EDC)