R Notebook

Mathematics Behind the Set Game

This writeup demonstrates an approach to represent various data structures for the Set game.

First we need a deck of cards.

deck = t(t(c(1:81)))
head(deck)

     [,1]
[1,]    1
[2,]    2
[3,]    3
[4,]    4
[5,]    5
[6,]    6

deck = data.frame(deck)

Each of the four attributes: number, color, shape, and texture will be represented by a number from 1 to 3.

This approach facilitates using modulo arithmetic to find the combinations of valid sets. A valid set is where each attribute sum modulo 3 is equal to 0.

onetwothree = c(1:3)
numbers = onetwothree
color = onetwothree
shapes = onetwothree
textures = onetwothree
# generate the 81 unique cards
i = 0
for (n in numbers) {
    for (c in color) {
        for (s in shapes) {
            for (t in textures) {
                i = i + 1
                deck[i,2] = n
                deck[i,3] = c
                deck[i,4] = s
                deck[i,5] = t 
                }
            }
        }
    }
colnames(deck) = c("deckID","number","color","shape","texture")
    
head(deck)

We now randomly pick a group of 12 cards. This uses the R function sample, which, by default, selects a sample without replacement.

Once we have the 12 cards, generate all combinations of 12 cards taken 3 at a time with the combn function.

These are then evaluated with modulo 3 arithmetic to find the actual valid sets.

# select our 12
pick12 = sample(deck[,1],12)
picks = deck[pick12,]
picks

allSets = combn(picks[,1],3)
dim(allSets)

[1]   3 220

totalSets = ncol(allSets)
for (i in 1:totalSets) {
    set = deck[allSets[,i],]
    #print (set)
    #print (sum(set[,2]))
    #print (sum(set[,3]))      
    isSet = (sum(set[,2]) %% 3 == 0) & (sum(set[,3]) %% 3 == 0) & 
        (sum(set[,4]) %% 3 == 0) & (sum(set[,5]) %% 3 == 0)
    if (isSet) {
        #print (i)
        print (deck[allSets[,i],])
    }
}

Translate this to text.

Colors = c(" Red"," Green"," Blue")
Shapes = c(" Oval"," Diamond"," Sqiggle")
Textures = c(" Solid"," Shaded"," Open")