In the card game Set, each card features a number of shapes, with four attributes:
Number: The number of shapes is 1, 2, or 3.
Color: Each shape is red, purple, or green.
Shape: Each shape is oval, diamond, or squiggle.
Shading: Each shape is hollow, shaded, or solid.
There is exactly one card for each possible combination of attributes.
In the game, several of the cards are dealt out, and the goal is to find a set. A set is formed by three cards, where for each attribute, either all three cards are the same, or all three cards are different.
So, I was wondering how many UNIQUE sets are there and please enter a text explanation. My first answer is $4^4=256$ but I'm not sure if I'm overcounting anything.
P.S. There are $3^4=81$ cards in a complete deck of Set. (Notice that the answer isn't undefined because there is only ONE card for each possible combination of attributes)
by at three sets are the same your talking about the same feature like same color?