Students' Questions, Conjectures, Etc:

About the Game SET

Question: (S.B.) How many cards are in the deck? (If all the possible cards described at setgame.com are in the deck and there are no duplicates)

Conjecture: (D.C.) Two cards define a SET.

Question: (C.C. and A.O.) Do 50-50’s count? Do 75-25’s count?

Question: (S.J. and S.B.) How much would 1 missing variable reduce SET possibilities?

Conjecture: (A.C.) If there aren’t any purple cards, then all the SETs have to be of one color.

Question: (D.C. and A.C.) How can you prove that there are no SETs in the array of cards you have?


Question: (T.R. and L.L.): Are all the cards in the deck to be used to make SETs with no cards being left over?

Conjecture: (T.R. and L.L.) If the answer to the above question is yes, and we have 9 remaining cards, there are 3 SETs left.

Conjecture: (T.R. and L.L.) After testing by continuing to look for SETs within the remaining 9 cards, we found the answer to the previous question to be no and the previous conjecture to be false

Statement: (T.R. and L.L.) The two previous conjectures are questionable because it can depend on the ways you make your SETs

Question: (T.C.) Does it matter if you (mistakenly, of course) take out three cards that are not a SET? Does that mess up the other SETs?

Conjecture: (T.R. and S.B.) The number of cards remaining at the end of a standard SET game depends on how the sets are arranged together.

Conjecture: (T.R. and S.B.) The number of cards left will always be 0 (mod 3).

Question: (T.R. and S.B.) Sometimes we found ourselves sorting out the cards. how does this help?

Question: (S.J. and L.L.) If all cards were used to make SETs, would the number of (0,4)’s, (1,3)’s, (2,2)’s, and (3,1)’s be even?

Conjecture: (S.J. and L.L.) Even if a whole deck was used, there couldn’t be an even number of SET’s per combination, since 4 doesn’t divide into 27 evenly. (4 combinations, 27 SETs)

Question: (S.J. and L.L.) What’s the least number of cards you can have left over?

Question: (A.C.) Is there a pattern to finding a set? Like first look for shape, then shading, color, and number.

Conjecture: (A.O.) When there are six cards left and only two colors there could be or could not be a set or two.

Question: (D. C.) If all colors, shapes, numbers, and shadings are present in a 12-card group, must there be a set within that group?

Question: (C. C.) Do psychology and perception have an effect on which types of sets (50-50, 75-25, etc.) seem to appear more within a game?

Data from S.J. and L.L - For SETs in a single game played:

# of Features All the Same	# of Features All Different	# of Occurrences
0	4	7
1	3	8
2	2	9
3	1	1

Question: (D. C.) What is the most efficient possible algorithm that can be used to find a SET?

Question: (D. C.) How many different ways are there to divide the deck into 27 SETS?

Question: (D. C.) What types of SETS are mostly likely to appear in 12 random cards?

Question: (D. C.) What is the probability that a given 12 cards will be missing a feature?

Conjecture: (D. C.) You can represent SET cards in 4 by 3 matrix.
The sum across the rows is 3, 6 or 9 to get a SET.

Proposed Definition: (D. C.) Given the representation of SET cards by 4-tuples where each component is either 1, 2, or 3; the numbers coding each of the three possibilities within each of the 4 features of the card: Define a binary operation * on the collection of SET cards by writing a*b=the SET card determined by card a and card b.

Conjecture: (A. O.) For the proposed binary operation above, the cancellation property holds. (This means that if a*b=a*c then b=c.)

Questions and Conjecture: (A. C.) Could two cards and a different two cards need the same card (to make a SET)? Two different sets of two could need the same card. How many pairs of two could need the same card?

Question: (A.C & L. L.) How would the probability of taking a card from the deck to make a SET change if a card in the deck can join with more than one pair to complete a SET?

Conjecture: (L. L.) A pair of cards, no matter what they are, always has potential to be part of a set.

Conjecture: (D. C.) Assuming that no sets are present in a collection of n cards and none have yet been removed, the probability that the next card will form a set is (nC2)/(81-n). (nC2 means the number of different pairs (2 objects, no particular order) of objects that can be chosen from among n objects.

Conjecture: (D. C.) Assuming the conditions above, there is always a set within 13 cards.

Conjecture: (D. C.) If there are 3 cards already laid out, any 2 cards out of the 3 cards have a 3/78 chance of becoming a set with the 4th card turned over and/or any 2 of those 3 original cards are guaranteed to make a set when put together with a matching card.

SOME SET PROBLEMS BASED ON THE CONJECTURES AND QUESTIONS ABOVE

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