Recall that the goal of the game of SET is to score points by finding SETs; groups of three cards where each of four card features is the same or different. Recall that the features are:
a) Number: one, two, or three
b) Color: red, green, or purple
c) Shape: oval, diamond, or squiggle
d) Texture: hollow, shaded, or filled
Since there are three values for each of the four card features, there are 3^4 = 81 cards in a box of SET.
For the sake of analysis, i.e., bringing some mathematical tools to bear, let's assign a number to each card feature:
a) Number: one=0, two=1, three=2
b) Color: red=0, green=1, purple=2
c) Shape: oval=0, diamond=1, squiggle=2
d) Texture: hollow=0, shaded=1, filled=2
Since each number above is 0, 1, or 2, we can consider them to be integers mod 3, i.e., elements of the group (Z3,+). Remember that this is the set {0, 1, 2} where the addition is done like the numbers are on a clock. For a demo, follow this link, set modulo equal to 3 and plus equal to 1 or 2. The arrows tell you what results when you add 1 or 2 to a number on the clock. The demo posted on a previous assignment is also helpful.
Exercise 1. If a feature present on three cards is all the same or all different, what do you notice about the sum of the associated numbers (modulo 3)?
For the sake of simplicity (for the moment), let's focus on just two card features (which two don't matter). So we're only considering 3^2 = 9 total cards.
