As a result of playing the game SET, the year 2004 Mount Holyoke College reSEARCH team posed the following problems and questions. As of June 29, the team was in the process of clarifying and answering these questions. Additional work resulting from attempts to solve the problems appear in red.

1. Are any of the cards in a SET deck duplicates? (G.A.)

2. How many cards are there in one SET deck? (J.M.)

3. How many of each color/shading/shape/number are in the deck? (M.W.)

4. How many cards with ovals are there in the deck? (G.A.)

Conjecture: There are 3^3=27 oval cards in the SET deck. (Conjecture, evidence, and argument given by K.V., M.W., T.K., & T.A.)

Conjecture: There are 3^4=81 cards in the SET deck. (Argument also given by K.V. M.W., T.K., & T.A.)

5. How many SETs can you make with 3 red cards, 3 purple cards and 3 green? Can you actually make a SET with these cards? Is there enough information given? Why or why not? (J.G.) See conjecture (J.G & G.A.) below.

6. How many SETs are there in the whole deck? (T.A., K.V., M.W.)

7. How many SETs are possible in an array of 24 cards, assuming that one card can be used for 2 SETs? (T.K.) See conjecture (M.W. & T.K.) below.

What is the probability?

8. How likely is it to have 4 SETs at the same time with no card in 2 sets? (G.A.)

9. What is the probability of finding a SET, where the shading, number, color and shape are all different? (K.V.)

10. What is the probability of finding a SET in an array of 15 cards? (J.G.)

11. If there are 2 different SETs in a deck of 21 cards, what is the probability of you picking, at random, a red card? (J.G.)

12. If there are duplicate cards, what is the probability of having an exact pair on the table at the same time? (G.A.)

Is it possible?

13. Is it possible to have an array of 12 cards in which there is no SET? (M.W.)

14. Is there a possibility a SET can consist of four or more cards? (T.K.)

15. Why must there always be 12 cards out to find a SET? (K.V.)

16. What is the significance of the number 3 in the game of SET? (T.A.)

17. Why are there 3 of each characteristic rather than 4 or 5? (T.K.)

Team members also posed the following conjectures:

1. If there are no duplicates, there are 27 cards with ovals. (G.A.)

2. The reason that there are always 12 cards played is related to there being 4 characteristics, 3 of each of them. (K.V.)

Above are some of the representations used by students to establish their conjectures related to problems #2 - 4.

Above is a conjecture concerning problem #5. Evidence given for the possibility of no SETS and 12 SETs, with a continuing investigation to see if there might be more than 12 SETs among the nine cards. All due to J.G. & G.A.

A conjecture (M.W. & T.K.) based on problem #7