Mathematics and the game "Set"

Whoever finds a set here will get a chocolate bar from me.

Seth - a brilliant game in which we were cut about 5 years ago. Screams, screams, photographing combinations.

It is written in the rules of the game that the geneticist Marsha Falco invented it in 1991, making notes during a 1974 study of epilepsy in German Shepherd Dogs. For those who have a brain rather exhausted by mathematics, after some time there is a suspicion that there are some echoes with planimetry and direct lines through points. (For two given cards, there is one and only one card that is included in one set with them.)


Marsha Falco as if asks: “Well, didn't you find the set?”
')

Remember the rules

Set is a card game. All maps have four parameters, each of which takes three values ​​(total 3 x 3 × 3 × 3 = 81 maps).



The types and values ​​of the parameters are as follows:

• shape :: = ellipse | rhombus | "Snot"
• color :: = red | green | Violet
• fill :: = white | striped | solid
• amount :: = 1 | 2 | 3

The goal of the game is to find special combinations of three cards. Three cards are called “set” if for each of the four attributes of the card either all are the same or all are different.



In other words, it can be said that three cards will not make a set if two cards have one parameter value, and the third one another. You can see that for any two cards there will always be a third (and the only one) with which they will be set.

Course of the game: the presenter puts 12 cards on the table. When someone finds a set, he shouts “Set!” And then calmly takes the components of the set card. If there are no set cards in the laid out cards (most likely, it only seems that they are not), the moderator lays out three more cards.

The maximum number of cards without a set is 20. The round continues until the deck ends. Wins the one who collected more sets.

Mathematicians hurry up and presented a combination of 20 cards. Whoever considers himself Chuck Norris, can forget this picture and try to collect “solitaire” without a set on his own.
Or check, and suddenly there is still a "set"?

20 cards without set

It is convenient to check that there is no “color set”.



The same maps, but the location shows that carried sets by the "fill".



In count.



According to the figures.



No set for the difference of signs.

Open unsolved problem of mathematics

How many cards can you put out to the maximum so as not to have a single “set”? The symptom has three meanings.

with 1 "sign" - 2 cards
2 signs - 4 cards
3 signs - 9 cards
4 signs - 20 cards
5 signs - 45 cards
6 signs - 112 cards
7 signs - xs

And when "n → ∞"?

Video

Game Creator:


Alexey Savvateev spontaneously talks about Seth:

Articles

• ON THE COMPLEXITY OF THE GAME OF SET
• SET Probabilities Revisited
• THE CARD GAME SET
• The Mathematics of the Card Game Set