Editing Set (card game) (section)

== Basic combinatorics of ''Set'' ==
{{Set_isomorphic_cards.svg}}
* Given any two cards, there is exactly one card that forms a set with those two cards. Therefore, the probability of producing a Set from 3 randomly drawn cards from a complete deck is 1/79.
* A [[cap set]] is a mathematical structure describing a Set layout in which no set may be taken. The largest group of cards that can be put together without creating a set is 20, proven in 1971 (cap sets were studied before the game).<ref>{{Citation |last=Hill |first=R. |title=On Pellegrino's 20-Caps in S4, 3 |date=1983-01-01 |url=https://www.sciencedirect.com/science/article/pii/S030402080873322X |work=North-Holland Mathematics Studies |volume=78 |pages=433–447 |editor-last=Barlotti |editor-first=A. |access-date=2023-12-16 |series=Combinatorics '81 in honour of Beniamino Segre |publisher=North-Holland |doi=10.1016/S0304-0208(08)73322-X |isbn=978-0-444-86546-5 |editor2-last=Ceccherini |editor2-first=P. V. |editor3-last=Tallini |editor3-first=G.}}</ref><ref>{{citation|last=Edel|first=Yves|doi=10.1023/A:1027365901231|issue=1|journal=Designs, Codes and Cryptography|mr=2031694|pages=5–14|title=Extensions of generalized product caps|volume=31|year=2004|s2cid=10138398}}.</ref><ref>{{Cite web|url=http://www.math.rutgers.edu/~maclagan/papers/set.pdf|title=The Card Game Set|author=Benjamin Lent Davis and [[Diane Maclagan]]|url-status=dead|archive-url=https://web.archive.org/web/20130605073741/http://www.math.rutgers.edu/~maclagan/papers/set.pdf|archive-date=June 5, 2013}}</ref> Such a group is called a maximal cap set {{OEIS|A090245}}. [[Donald Knuth]] found in 2001 that there are 682344 such cap sets of size 20 for the 81-card version of Set; under affine transformations on 4-dimensional finite space, they all reduce to essentially one cap set.
* There are <math>\textstyle\frac{{81 \choose 2}}{3} = \frac{81 \times 80}{2 \times 3} = 1080</math> unique sets.
* The probability that a set will have <math>d</math> features different and <math>4 - d</math> features the same is <math>\textstyle\frac{{4 \choose d}2^d}{80}</math>. (Note: The case where ''d''&nbsp;=&nbsp;0 is impossible, since no two cards are identical.) Thus, 10% of possible sets differ in one feature, 30% in two features, 40% in three features, and 20% in all four features.
<!-- * When a complete deck of 81 Set cards is partitioned into two piles of size n and (81-n), the sum of the number of Sets that can be made using only cards in the first pile plus the number of Sets that can be made using only cards in the second pile has a maximum given by <math>\frac{{81 \choose 2}}{3} - \frac{n(81-n)}{2}</math>. -->
* The number of different 12-card deals is <math>\textstyle{81 \choose 12} = \frac{81!}{12! 69!} = 70\,724\,320\,184\,700 \approx 7.07 \times 10^{13}</math>.
* The odds against there being no ''Set'' in 12 cards when playing a game of Set start off at 30:1 for the first round. Then they quickly fall, and after about the 4th round they are 14:1 and for the next 20 rounds, they slowly fall towards 13:1. So for most of the rounds played, the odds are between 14:1 and 13:1.<ref name="Revisited">{{Cite web|url=http://henrikwarne.com/2011/09/30/set-probabilities-revisited/|title=SET Probabilities Revisited|date=30 September 2011|access-date=4 October 2011|archive-date=10 December 2011|archive-url=https://web.archive.org/web/20111210084923/http://henrikwarne.com/2011/09/30/set-probabilities-revisited/|url-status=live}}</ref>
* The odds against there being no Set in 15 cards ''when playing a game'' are 88:1.<ref name="Revisited" /> (This is different from the odds against there being no Set in ''any'' 15 cards (which is 2700:1) since during play, 15 cards are only shown when a group of 12 cards has no Set.)
* Around 30% of all games always have a Set among the 12 cards, and thus never need to go to 15 cards.<ref>{{Cite web|date=2011-09-30|title=SET® Probabilities Revisited|url=https://henrikwarne.com/2011/09/30/set-probabilities-revisited/|access-date=2022-02-07|website=Henrik Warne's blog|language=en|archive-date=2022-02-07|archive-url=https://web.archive.org/web/20220207053109/https://henrikwarne.com/2011/09/30/set-probabilities-revisited/|url-status=live}}</ref>
* The maximum number of Sets for 12 cards is 14.<ref>{{Cite web|date=2025-01-25|title=The Maximum Number of Sets for 12 Cards is 14|url=https://arxiv.org/abs/2501.12565|url-status=live}}</ref>
* The average number of available Sets among 12 cards is <math>\textstyle{12 \choose 3} \cdot \frac{1}{79} \approx 2.78</math> and among 15 cards <math>\textstyle{15 \choose 3} \cdot \frac{1}{79} \approx 5.76</math>. However, in play the numbers are smaller.
* If there were 26 sets picked from the deck, the last three cards would necessarily form another 27th set.