Questions, Problems, & Conjectures about SET

From the 2005 ReSEARCH Group

1. In the game SET, there are 9 cards for each pattern. (L.V. & R.F.)

2. (Refinement of #1) In the game SET, once 2 of the features (characteristics) are given, there are exactly 9 cards having those features. (C.F.)

3. There are exactly 81 cards in a perfect SET deck. (C.F.)

4. Any collection of 13 cards is guaranteed to contain a SET. (R.F.)

5. There is a collection of 13 cards that contains no SET, which is a contradiction to #4 above. (R.B.)

6. What is the probability of getting a SET on the 12th card turned over if there is no SET among the previous 11 cards turned over? (R.F.)

7. The probability of getting a SET on the 12th card turned over, if there is no SET among the previous 11 cards turned over, is 12/81 = 4/27. (R.F.)

8. Question: For any given SET card, how many different SETs is it a member of? The question was asked by and answered by R.B. and C.F. as a subproblem to help solve the problem of determining the probability that the 5th card turned over in a SET game forms a SET, given the the first 4 cards turned over do not contain a SET.

9. How many pairs of cards are there in a collection of 14 cards? The question was asked by J.M. as a question related to the problem of finding a collection of 14 cards that contains no SET. The question was answered by L.M.

10. How many pairs of cards are there in a collection of:
a) 3 cards, b) 4 cards, c) 5 cards, d) 6 cards, e) 7 cards, f) 8 cards, g) 9 cards, h) 16 cards? Answers were determined by L.M., L.V., & ?

11. For any SET card, how many different SETs is that card a member of? (C.F. & R.B.) Answers given by C.F., R.B., & L.M.

12. Conjecture. The maximum number of cards containing no SET is 17. (R.B. & C.F.)