What is the math behind the game Spot It? I just purchased the game Spot It. As per this site, the structure of the game is as follows:


  Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 different symbols through the deck. The most fascinating feature of this game is any two cards selected will always have ONE (and only one) matching symbol to be found on both cards.


Is there a formula you can use to create a derivative of this game with different numbers of symbols displayed on each card.

Assuming the following variables:


S = total number of symbols
C = total number of cards
N = number of symbols per card


Can you mathematically demonstrate the minimum number of cards (C) and symbols (S) you need based on the number of symbols per card (N)?
The celebrated Ray-Chaudhuri–Wilson theorem states that $C \leq S$, contradicting your numbers.

An almost matching construction is as follows. Pick some prime number $n$. Our universe, of size $n^2+n+1$, consists of pairs of numbers in $\{0,\ldots,n-1\}$ plus $n+1$ singletons $\{0,1,\ldots,n-1,\infty\}$ ("points at infinity"). For each $0 \leq a \leq n-1$ and $0 \leq b \leq n-1$ we will have a card of size $n+1$ containing the pairs $\{(x,ax+b \mod{n})\}$ and the singleton $a$. There are also $n$ special cards, for each $0 \leq c \leq n-1$, containing the pairs $\{(c,x)\}$ and the singleton $\infty$. One super special card contains all $n+1$ singletons.

Clearly two cards with the same $a$ intersect only at the singleton. Two cards with different $a$s intersect at the unique solution to $a_1x+b_1 = a_2x+b_2 \pmod{n}$. Two special cards intersect only at the singleton, and a normal and a special card intersect at $(c,ac+b)$. Finally, the super special card intersects the rest at a singleton.

In total, we have $n^2+n+1$ cards and symbols, each card containing $n+1$ symbols, and two cards intersecting at exactly one symbol. In your case $n=7$ and so the number of cards and symbols should be $7^2+7+1 = 57$.
N squared minus N + 1 is the correct formula for calculating both the number of images and the number of cards.  Here is the specification for generating the cards themselves:
Below are the specifications for generating each card of Spot-it.  Although the commercial Spot-it game contains 8 symbols per card, these specifications work for any desired number of symbols/card.  The rules are that every card contains one and only one symbol matching every other card.  Also, all symbols have an equal probability of being the matching symbol.

Let N = number of symbols per card
Let C = the total number of symbols as well as the total number of cards to be generated.
Then C = N2 – N + 1

Generate a matrix with N columns and C rows
Column 1:  Symbol 1 for N rows
            Symbol 2 for N-1 rows
                   .
                             .
                   .
            Symbol N for N-1 rows
Column 2:  Row 1 = Symbol (Column #)
            Row 2 = Symbol N+1
            Row 3 = Symbol 2N
                      Row 4 = Symbol 3N-1
                      Row 5 = Symbol 4N-2
                      Etc. through row 2N-1
             Repeat above rows N-2 times
Column 3:  Row 1:  Symbol (Column #)
             Row 2:  Previous column Row 2 plus 1
                      Row 3:  Previous column Row 3 plus 1
             Etc. through row 2N-1
             Repeat above rows N-2 times
Column 4:  Same as Column 3
       .
       .
       .
Column N

The Spot-it Cards are the Rows