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Question (brief introduction to the game Set is given after the question)

When a game of Set gets to a point that there are only three cards left on the table, and all other cards were already removed as part of a Set, do these three cards always form a Set?

The game of Set can be described by a finite four-dimensional affine space with each line containing three points. Every Set then is in correspondence with one of the lines in this space.

When fiddling around with a two-dimensional case (i.e. an affine space consisting of nine points), the question seems true. Also, by experience of playing the game, I'm inclined to say the question is true in general. How would one go about proving this?

Brief introduction to the game Set (may be skipped if familiar)

The game Set is a card game consisting of 81 cards, all with a different combination of the following properties:

  • Shapes: Diamond, Oval, Squiggle
  • Color: Purple, Red, Green
  • Number: 1, 2, 3
  • Filling: Open, Lined, Solid

The objective is to find three cards in the cards dealt on the table such that for all four properties either all three cards have the same occurrence of this property (e.g. three red cards), or they have the three different occurrences (e.g. one diamond, one oval and one squiggle). These triplets are called Sets.

When someone has found a valid Set, these three cards are removed from the table and three new cards are added. Initially the game starts with 12 cards on the table, but sometimes it is required to add more (triplets of) cards, since there is a possibility that for $n<21$ cards on the table, there is no Set among them.

The game ends when there are no more Sets among the last cards on the table. It is entirely possible (and most likely) the game will end before all cards are removed, leaving behind some cards on the table.

konewka
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1 Answers1

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I'll just note that it is possible for a game to end with more than 3 cards and no sets on the table, but if there are three cards left at the end then indeed they must form a set.

Under the identification of a set deck with $\mathbb{F}_3^4$ (which it sounds like you are familiar with) three cards form a set iff they satisfy $x+y+z =0$. Moreover, the sum of all $81$ cards is also zero. So every time you remove a set the sum of the remaining cards is also $0$. Hence if you are left with three cards after removing a bunch of sets they must sum to $0$ and therefore form a set.

Nate
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