The game SET is a card game where you each card has four attributes: Color, number, shape, and shading. Each attribute has 3 types: each card is purple, red or green; oval, diamond, or squiggle; has 1,2 or 3 items on it; and is clear, lightly shaded, or filled in. A set consists of three cards that satisfy the following condition: in each attribute they are either all the same or all different.
The total number of cards in the game is 3^4 since there are 4 different attributes and each attribute has 3 types. One of the most interesting questions about the game is, what is the maximum number of cards you can lay down on a table and have no SET whatsoever.
It turns out the answer is 20, and there are reasons for this that come from linear algebra! Using linear algebra in 4-dimensions, where vectors have components being integers mod 3, one can prove that no matter what 21 cards you lay down on a table, there *has* to be a SET.
The image shows a glimpse of 12 of the cards in a 20 card table that has no SET.
Mohamed Omar
The game SET is a card game where you each card has four attributes: Color, number, shape, and shading. Each attribute has 3 types: each card is purple, red or green; oval, diamond, or squiggle; has 1,2 or 3 items on it; and is clear, lightly shaded, or filled in. A set consists of three cards that satisfy the following condition: in each attribute they are either all the same or all different.
The total number of cards in the game is 3^4 since there are 4 different attributes and each attribute has 3 types. One of the most interesting questions about the game is, what is the maximum number of cards you can lay down on a table and have no SET whatsoever.
It turns out the answer is 20, and there are reasons for this that come from linear algebra! Using linear algebra in 4-dimensions, where vectors have components being integers mod 3, one can prove that no matter what 21 cards you lay down on a table, there *has* to be a SET.
The image shows a glimpse of 12 of the cards in a 20 card table that has no SET.
4 years ago | [YT] | 23