What is Set?

Set is a card game where you try to find "Sets". The more sets you find, the better. First, we need to define a set. A set consists of 3 cards, and must have either every property in common or every property different. So, two cards in a set can't have one property different than the third card in the set. The properties are Color, Shading, Shape, and Number. The Colors can be Purple, Green, and Red. The Shading can be solid, stripes, or blank. The Shape can be an S, a diamond, or an oval. The Number can be 1, 2, or 3. It's a bit easier to explain by using examples.

Shape- All 3 are different
Color- All 3 are different
Shading- All 3 are different
Number- All are the same; they have 2 numbers

Shape- All are the same; oval
Color- All are the same; red
Shading- All are the same; blank
Number- All are different

Shape- All 3 are different
Color- All 3 are different
Shading- All are the same; stripes
Number- All 3 are different


Shape- All 3 are different
Color- All are the same; they are purple
Shading- All 3 are different
Number- All are the same; they have 3 numbers


Shape- All 3 are different
Color- All 3 are different
Shading- All 3 are different
Number- All are the same; they have 3 numbers

So, now that you understand what a set is, we can play the game. Here's an example game below. There are 6 possible sets. The answers are further down.

Here's all of the possible sets:

2-3-8 5-3-1 5-6-8 1-2-9 3-9-6 8-10-11

I want to try and prove what is the largest amount of sets you can get from 12 cards, if all of the cards are different. 

2 of the 4 characteristics can be represented, with the 2 being shading and number. The shape and color stays the same.The first idea that comes to mind is finding the maximum amount with 9 cards; that fits nicely with the 3 different characteristics in each group. I think it would look something like this:









So, if we have these 9 cards, there would be 12 sets. Seems like we will have 12 sets when we have 12 cards, right?
Well, 12 cards will be harder to account for. There aren't 4 characteristics in any of the groups, making it difficult to choose which cards to use for the remaining 3 spots.
And, what's worse, we can't really just add on something like this, because it wouldn't work well with the red.

So, If this were added to the set, you would only be adding 1 set to the group, which is all of the 3 greens together. You can't mix the red and green together, because there will always be 2 red cards and 1 green card, or 1 red card and 2 green cards, which both can't make sets.

I think there is one better option than this;

I believe that this is the best thing you can add on to the 9 cards, it creates 2 extra sets instead of 1. I'll show how the sets are made.

This should be the best 12 card game.


The 2 new sets that would be created are these:

These sets kind of solve the problem of having only red and green cards. We add on a purple card with 3 shapes instead of 2, because if it was purple with 2 shapes, it would only create 1 set.

Now, we can create cards each with 1, 2, and 3 shapes, and different colors.

The reason why we don't have a set with the 1st card being a striped red diamond and the 2nd card being a solid green card is because we don't have enough cards to add on; we only have 3 extra cards to add, even though we really want to add 4. So, if we had a 13 card game, we would be able to get an extra set added on.

So, I believe that a 12 card game can have a maximum of 11 possible sets, if all of the cards are different. 

So, it looks like we're all done!

Well, there are a few more things we could do...

What if a set could have cards be the same as each other? What if every single card was the same??? How many sets could be made?


Here's an example of having every single card be the same:

I know. It looks absolutely crazy hard to understand.
No sarcasm whatsoever at all.





This is actually pretty easy; there are 12 cards to choose from at the beginning, and after selecting your first card, you have 11 cards to choose from, and 10 cards which could represent the last card.

So, really what we're doing is 12*11*10, or just multiplying together all the possibilities of picking the 12, 11, and 10 cards available, resulting with 1320 possible combinations.

There was actually a formula created to do this easier, called the permutation formula. 

This is the formula, where N is the number of cards and R is the number of things you are choosing from.

P(n,r) = n!/(n-r)!

So, if N is 12 and R is 3,
just put it into the formula, and solve.


P(12,3) = (12!)/((12-3)!)

P(12,3) = (12!)/(9!)

P(12,3) = (12*11*10*9!)/(9!)

P(12,3) = 12*11*10

P(12,3) = 1320

Hm... seems oddly familiar, right?

Really what this equation was doing was setting up the 12*11*10.

It uses the "!" factorial to do this, and does it well. 

Basically, it takes the top number 12, and says that it should multiply numbers 12, 11, 10, 9, 8, etc.

So, the 3 tells it to stop at 10, so that it only multiplies 12*11*10, which is what you want.

 Yes we're done.