## Teaching Notes for the

Game of Set Problem

#### Introducing the Project

Day 1 Start by explaining the rules and letting students play the game a few
times.

If students have difficulty finding sets, have them play an easier version with only
27 cards and one of the attributes fixed (e.g., use only red cards).

Let students practice locating Sets, explaining why they are Sets, and proving
that no Sets are among the cards on the table. Encourage students to share the
strategies that they use looking for Sets, any observations they have, or any questions
that emerge. Seek to make their questions a focus of some of the research for them,
their group, or the class.

For some of the problems, it is helpful to come up with a simple notation which
can be shared by all the students and used in discussing the game. The more simple
the notation is, the more useful it will be in the process of the solution. A possible
notation can be based on assigning a number from 1 to 3 to each value of an
attribute and then fixing the order in which these attributes are described. For
example, a card can be represented by four numbers listing the shape of its
forms, then the number of the shapes, then the color, and at last the shading.
{1,2,3,2} might be written for the card with two forms of shape # 1, with color
#3 and shading #2. This notation makes more obvious the equality of the
attributes. For example, if instead of three different shadings we use three
different background colors, the game will not change in any meaningful way.
Also, if we consider the attributes in a different order, nothing essential will
change.

Day 2 Students can work on the warm up problems. If students do not have a
background in combinatorics, the first question should take some time and careful
analysis (including the study of simpler cases with fewer attributes). For more
experienced students, the main question that asks for the maximum number of Sets
within 9 cards can be used as a warm up for the question about the maximum Set-less
pile. Working with a deck of cards, students can try to construct Set-rich groups. The
teams can share their strategies for assembling their groups. This problem
introduces the notion of establishing upper and lower limits for a problem.
Each time an example is constructed with a higher number of Sets, that
establishes a new lower limit (this is a case of proof by example!). A more formal
proof is required to show that a certain number of Sets is not attainable.
For example, with 9 cards, there are only _{9}C_{3} = 84 different subsets of 3
cards. So 84 is an upper limit for the number of Sets. This calculation is a
considerable overcount, because it does not account for the fact that two different
Sets cannot have two cards in common. However, it is a good start. The
narrowing of the gap between lower and upper bounds for an answer is a main
theme for this project and is a common approach to many combinatorics
questions.

Day 3 Students can work in groups on some of the warm up problems and then
present and compare their solutions and methods. Basic notions of the uniqueness of
the third card needed to complete a Set and of the use of multiplication and division
to count combinations should be established during the warm up explorations. For
students new to combinatorics, the warm up problems will be challenging
undertakings in their own right.

Once students have successfully worked on some of the warm-up problems,
they can choose one of the four main questions (or a new question that has
emerged during class) to investigate independently. You might want to let the
students who chose to work on the maximum Set-less pile know that this
problem was only recently solved analytically and remind them that partial
answers, such as “This number must be less (or greater) than ...” are also very
valuable.

#### Working on the Project

You might devote periodic class meetings to independent or small group work on the
project and also invite students to present updates on their findings to the
class.