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Replying to Dan Brumleve's recommendation of the card game "Set" as an answer because my comment was exceeding the maximum allowed size.

The card game "Set" is most analogous to playing tic-tac-toe on a 4-dimensional lattice of size $3 \times 3\times 3\times 3$, allowing for lines to wrap around in that 4-space.

As Douglas Zare described it, the "game asks you to recognize lines in affine 4-space over the field with 3 elements." There are $3^4=81$ cards, with each card showing a design that can be described by 4 attributes, with each attribute containing 3 elements: 3 colors, 3 shapes, 3 levels of shading (outlined, striped, solid), and 3 cardinalities (1,2, or 3).

Twelve cards are initially dealt with players competing to find a grouping of 3 cards such that their attributes are collinear in the 4-d space: either all the same or all different. Thus each line in affine space can also be described by the vector $v\in${$-1, 0,+1$}$^4$ (but excluding ${0}^4$) and a representative member of that grouping. Each card can be also be seen as describing a permutation on the group of the 81 cards.

One quick question that comes out of this is

• Are 12 cards sufficient to guarantee that the cards dealt contains such a collinear grouping?

The answer to that is no. If the people playing concur that the 12 cards initally dealt does not contain 3-points in the 4-d lattice that are collinear, then 3 more cards are dealt, etc.

• What is the minimum number of cards that must be selected to guarantee that that it contains a collinear group of three? The answer to that (21) must be one more than the answer to

What is the maximum number of cards that can be played which do not contain a collinear group of three? (ans=20)

The card game misuses the mathematical term "set" in its name and in its directions for playing the game, since it asks the players to yell out "set!" when they find such a collinear grouping in the 4-d lattice. It should be rightfully called "line", or perhaps most correctly "4-dimensional affine space line over the field with 3 elements". But yelling that out each time would certainly slow down playing the game. :)

Replying to Dan Brumleve's recommendation of the card game "Set" as an answer because my comment was exceeding the maximum allowed size.

The card game "Set" is most analogous to playing tic-tac-toe on a 4-dimensional lattice of size $3 \times 3\times 3\times 3$, allowing for lines to wrap around in that 4-space.

As Douglas Zare described it, the "game asks you to recognize lines in affine 4-space over the field with 3 elements." There are $3^4=81$ cards, with each card showing a design that can be described by 4 attributes, with each attribute containing 3 elements: 3 colors, 3 shapes, 3 levels of shading (outlined, striped, solid), and 3 cardinalities (1,2, or 3).

Twelve cards are initially dealt with players competing to find a grouping of 3 cards such that their attributes are collinear in the 4-d space: either all the same or all different. Thus each line in affine space can also be described by the vector $v\in${$-1, 0,+1$}$^4$ (but excluding ${0}^4$) and a representative member of that grouping. Each card can be also be seen as describing a permutation on the group of the 81 cards.

One quick question that comes out of this is

• Are 12 cards sufficient to guarantee that the cards dealt contains such a collinear grouping?

The answer to that is no. If the people playing concur that the 12 cards initally dealt does not contain 3-points in the 4-d lattice that are collinear, then 3 more cards are dealt, etc.

• What is the minimum number of cards that must be selected to guarantee that that it contains a collinear group of three? The answer to that (21) must be one more than the answer to

What is the maximum number of cards that can be played which do not contain a collinear group of three? (ans=20)

The card game misuses the mathematical term "set" in its name and in its directions for playing the game, since it asks the players to yell out "set!" when they find such a collinear grouping in the 4-d lattice. It should be rightfully called "line", or perhaps most correctly "4-dimensional affine space line over the field with 3 elements". But yelling that out each time would certainly slow down playing the game. :)