Questions, Problems, & Conjectures about SET

From the 2006 ReSEARCH Group

Questions/Conjectures from ReSEARCH Session 1:

1. Is it possible to have 12 cards containing no SET? (Aviva S.)

2. Conjecture: It is possible to have 12 cards containing no SET. (Counterexamples proposed by Lisa M. & Alicia L. and by Adrienne F. & Jane L)

3. What is the maximum number of SETs in 12 cards? ( A.S.)

4. Do all cards eventually end up in SETs? (Adrienne F. & Jane L)

Questions/Conjectures from ReSEARCH Session 3:

1. Conjecture: When you first start a SET game and you have just turned over the first two cards, the chance that the very next card will form a SET is 1 in 79. (Conjectured by many)

2. Conjecture: Suppose you are starting a SET game and you have turned over the first three cards, and they do not form a SET. The probability that the very next card turned over will form a SET with two of the other three cards is 3/78. (Conjectured by many)

3. Conjecture: Suppose you are starting a SET game and you have turned over the first four cards, and they do not form a SET. The probability that the very next card turned over will form a SET with some two of the other four cards is 6/77. (Conjectured by many)

Questions/Conjectures from ReSEARCH Session 5:

Proof is given that Conjecture 1 from Session 3 is true. (by many)

Questions/Conjectures from ReSEARCH Session 7:

SET

Question (Shelby H., Dary H., Alicia L - also noted and proven to have answer "No" by Lisa M., Jane L., and Nahid A.): If there are 3 cards could the fourth card complete 2 sets?
Answer: NO

Conjecture: If the first three cards do make a SET, then the probability that the fourth card also makes a SET is 0. (Aviva S.)

Proof that Conjecture 2 (from Session 3) is true given by Lisa M., Jane L., and Nahid A., by Aviva S., Adrienne F., and Cherelle H., and by Mariel M., Yasemin O., and Alicia C.

Examples to show that Conjecture 3 (also from Session 3) is false given by Lisa M., Jane L., and Nahid A., by Aviva S., Adrienne F., and Cherelle H., and by Mariel M., Yasemin O., and Alicia C.

Conjecture: The probability referred to in Conjecture 3 can be either 6/77 or 5/77, depending on which four cards are turned over. (Lisa M., Jane L., and Nahid A., by Aviva S., Adrienne F., and Cherelle H., and by Mariel M., Yasemin O., and Alicia C.)

Conjecture: When 4 cards have been turned over the probability that the 5th card will form a set with two of the first 4 cards can be 4/77. (Jane L.)

Question: Can the probability, again for Conjecture 3, be as low as 3/77?

Question: How many SETs can one card complete? (Alicia C.) Conjectured answers: 40 by Jane L. 80 by others.

Conjecture: If you have 12 cards down with no sets the 13th card will complete at least one set. (Yasemin O.)

Conjecture: If there are no sets up to 13th card, and there are no pairs that require the 3rd card in common, then the probability to make a SET is 1 by drawing the 13th card. (Yasemin O., Mariel M., and Alicia C.)

Conjecture: Having pairs that require the same 3rd card, or 3rd card in common, decreases the probability of getting a SET. (Yasemin O., Mariel M., and Alicia C.)

Questions/Conjectures from ReSEARCH Session 12: Progress Reports

Conjectures from the Maximizing SETs Group: Cherelle H., Lisa M., Dary H., & Yasemin O.:

Conjecture: You can pick nine cards that contain 12 SETs.
Conjecture: If you take out one card from the nine cards that contain 12 SETs, then we’ll have at most eight SETs.
Conjecture: If a tenth card is added to the nine cards that contain 12 SETs, then another SET wouldn’t be made.

Conjectures from the Simpler Form of SET Group: Jane L., Adrienne F., Alicia C., & Aviva S.:

Conjecture: In two-dimensional SET, after the first two cards are dealt, the chance of a SET on the third card is 1/7.
Conjecture: In two-dimensional SET, after the first three cards are dealt and there is no SET, the chance of a SET on the fourth card is 3/6 = 1/2.
Conjecture: In two-dimensional SET, after the first four cards are dealt and there is no SET, the chance of a SET on the fifth card is 5/5 = 1.
Conjecture: In two-dimensional SET, there is a maximum of two Sets in a group of five cards.
Conjecture: In two-dimensional SET, there is a maximum of three Sets in a group of six cards.
Conjecture: In two-dimensional SET, every card is contained in four SETs.
Conjecture: In four-dimensional SET, every card is contained in 40 SETs.

Conjectures from the SET Checker Group: Nahid A., Shelby H., Alicia L., & Mariel M.:

Conjecture: The following procedure can be used to check for a SET among three cards: Input the 4-tuple codes for the three cards into the columns labeled card#1, card#2, and card#3 as illustrated

Conjecture: Checking four cards for SETs can be done by performing the test in the previous conjecture on each of the four 3-card combinations of the four cards.

Conjecture: If the three cards form a SET, then the sum of all the coordinates of all three cards is a multiple of three. However, there areIf all four entries in column F are zero, then the three cards form a SET; otherwise, the three cards do not form a SET. (Shelby H. & Nahid A.)

Conjecture: If the sum of the four numerical entries in column F is zero, then the three cards form a SET, and, otherwise, the three cards do not form a SET. (Aviva S.)