Set (Pattern Card Game)
Any two cards determine a unique third — which is why the 81-card deck contains exactly 1,080 sets, by arithmetic rather than search.
Any two cards determine a unique third
Pick any two cards in the deck and exactly one other card completes a set with them. For each of the four attributes, if the two agree the third must match, and if they differ the third must take the remaining value. There is never a choice and never an ambiguity — which is the elegant fact the whole game rests on, and it is why "I can't find one" so often means "I am not looking at pairs".
It also makes the deck exactly countable without any searching. Eighty-one cards give 3,240 pairs; each pair belongs to exactly one set and each set contains three pairs, so there are 3,240 ÷ 3 = 1,080 sets in the deck. That is arithmetic rather than enumeration, though the tests behind this page do enumerate all 85,320 possible triples to confirm it comes to the same number.
How to use
- Pick three cards where each attribute is all-same or all-different.
- Attributes are number, shape, shading and colour.
- Take a valid set and three new cards replace it.
- Stuck? Look at pairs — any two cards imply exactly one third.
Frequently asked questions
What counts as a set?
Three cards where each of the four attributes — number, shape, shading and colour — is either the same on all three or different on all three. Two-and-one never works for any attribute. Three cards that are all red, all diamonds, all striped, with one, two and three symbols, is a valid set because every attribute is consistent.
Do two cards always have a third that completes them?
Yes, exactly one, and that is the elegant fact the game rests on. For each attribute, if the two cards agree the third must match, and if they differ the third must take the remaining value. There is never a choice and never an ambiguity, so every pair in the deck belongs to precisely one set.
How many sets are in the deck?
One thousand and eighty, and you can get there without searching. Eighty-one cards give 3,240 pairs; each pair belongs to exactly one set, and each set contains three pairs — so 3,240 divided by 3 gives 1,080. Brute-force enumeration of all 85,320 triples confirms the same number.
Why does the game deal three more cards sometimes?
Because occasionally twelve cards contain no set at all. Rather than reshuffle, the rules add three more and keep adding until a set exists. It is uncommon — only a small percentage of twelve-card layouts are setless — but it happens often enough that a rule is needed.
How do I get better at finding sets?
Work from pairs rather than scanning triples. Pick two cards, work out what the third must be for each attribute, and then look for that card — the unique-third property means you are searching for one specific thing rather than checking combinations. It is far faster than the way most beginners look.
What is the largest group of cards with no set?
Twenty. It is a known and non-obvious result — no collection of twenty-one cards from the deck can avoid containing a set, and collections of twenty that do exist are rare and structured. The question generalises to a well-studied problem in finite geometry called the cap set problem.
Is Set related to mathematics?
Closely. The deck is a four-dimensional space over three elements, and a set is exactly a line in that space — which is why any two points determine a unique third, since three points make a line when they sum to zero in each coordinate. The cap set problem, about the largest setless collection, is genuine research mathematics.
Why is the shading attribute hard to see?
Because it is the one people forget to check. Number, shape and colour are all obvious at a glance, but open, striped and solid look similar in peripheral vision, so most missed sets and most false calls come from shading. Checking it deliberately is a habit worth building.
Does the game get easier as it goes?
Slightly, because fewer cards remain in the deck and the layout thins out, but not as much as you would expect — with fewer cards on the table there are fewer sets available too. What genuinely changes is that the remaining cards become less varied, which makes some attributes stop discriminating.
Who invented Set?
Marsha Falco, a population geneticist, who devised it in 1974 while using symbol cards to encode genetic data in German Shepherds. She noticed her colleagues playing with the cards and turned it into a game, which was published commercially in 1991 and has won a great many awards since.
Can a set contain three identical cards?
Not in a real game, because every card in the deck is unique — there is exactly one of each of the 81 combinations. Three identical cards would satisfy the rule in the abstract, since every attribute would be all-same, but the situation cannot arise with a standard deck.
Does the game save my progress?
No. Everything runs in your browser, nothing is stored or uploaded, and closing the tab loses the current game. Each deal is shuffled freshly.
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