Math of the game SPOT IT!

Spot It! is a fast-paced matching card game. Each of the 55 cards in the deck features eight symbols, and there is always exactly one matching symbol between any two cards in the deck. Your goal is to be the quickest to find the match between two cards. There are a total of 57 different symbols throughout the deck. While playing the game with several family members in 2014, my son questioned how it was possible that any two cards always have exactly one match.

After spending some time analyzing the game, I came to the conclusion that the makers of Spot It! left out two cards! They could have made 57 cards out of the 57 symbols with 8 symbols on each card. I will try to explain the mathematics involved.

If there were 2 symbols on a card, then the deck would contain 3 cards and a total of 3 different symbols.
Let the symbols be A, B, and C.
Then there would be 3 cards: AB, AC, and BC.
Note that between any two of the cards, there is one and only one matching symbol.

If there were 3 symbols on a card, then the deck would contain 7 cards and a total of 7 different symbols.
Let the symbols be A, B, C, D, E, F, and G.
Then there would be 7 cards: ABC, ADE, AFG, BDF, CDG, CEF, and BEG.
Note that between any two of the cards, there is one and only one matching symbol.
Also, note that each symbol occurs on 3 cards.

If there were 4 symbols on a card, then the deck would contain 13 cards and a total of 13 different symbols.
Let the symbols be A, B, C, D, E, F, G, H, I, J, K, L, and M.
Then there would be 13 cards:
ABCD AEFG AHIJ AKLM
BEHK BFIL BGJM
CEIM CFJK CGHL
DEJL DFHM DGIK
Note that between any two of the cards, there is one and only one matching symbol.
Also, note that each symbol occurs on 4 cards.

I was able to create the 13 cards very easily by using a table similar to the one I use in solving logic puzzles.
I used the method of elimination to solve for the 13 unique cards where there was exactly 1 match between any two of them.
I used X's to eliminate the possibilities, and O's for the symbols on the cards.
Here is the table:

A B C D E F G H I J K L M

O O O O X X X X X X X X X

O X X X O O O X X X X X X

O X X X X X X O O O X X X

O X X X X X X X X X O O O

X O X X O X X O X X O X X

X O X X X O X X O X X O X

X O X X X X O X X O X X O

X X O X O X X X O X X X O

X X O X X O X X X O O X X

X X O X X X O O X X X O X

X X X O O X X X X O X O X

X X X O X O X O X X X X O

X X X O X X O X O X O X X

Now, let us jump to cards with 8 symbols on each card (which the makers of SPOT IT! used in their game).
If there were 8 symbols on a card, then the deck could contain 57 cards and a total of 57 different symbols.
Let the symbols be the numbers from 1 to 57.
Then the cards which contain the number 1 would be:
1 and the numbers from 2 to 8
1 and the numbers from 9 to 15
1 and the numbers from 16 to 22
1 and the numbers from 23 to 29
1 and the numbers from 30 to 36
1 and the numbers from 37 to 43
1 and the numbers from 44 to 50
1 and the numbers from 51 to 57
Note that between any two of the cards above, there is one and only one matching symbol -- the number 1.
These are the 8 cards that contain the symbol 1, and it will not occur on any of the other cards.

The next thing that I did was to create a table showing the number of symbols on a card and the corresponding maximum number of cards in that deck.
Note that the total number of times each symbol is used is the same as the number of symbols on each card.
Also, the total number of symbols needed throughout the deck is equal to the maximum number of cards for the deck.

# of symbols on a card # of cards in that deck

2 3

3 7

4 13

5 21

6 31

7 43

8 57

9 73

10 91

n n² - n + 1

Hopefully, you found a pattern in the table above for the number of cards needed for a deck:
Starting at 2 symbols on a card, there are 3 maximum cards for the deck.
Then for 3 symbols, add 4 to obtain 7 cards in the deck.
Then for 4 symbols, add 6 to the previous number to get 13 cards.
Then for 5 symbols, add 8 to the previous number to get 21 cards.
Then for 6 symbols, add 10 to the previous number to get 31 cards.
Then for 7 symbols, add 12 to the previous number to get 43 cards.
Then for 8 symbols (which is used in SPOT IT! ), add 14 to the previous number to get 57 cards.
Mathematically, the formula for the number of cards for n symbols on a card is given by n² - n + 1.

Again, I don't know why the makers of the game of SPOT IT! did not use 57 cards. And since two cards are missing, not all of the symbols are used equally! So, there is a greater chance of certain symbols being the match than other symbols! You could use this fact to your advantage in playing the game! Check the symbols that are used less frequently last! I do not have a deck of SPOT IT! or I would check to see which symbols are used less than 8 times, which is the maximum that any symbol could be used.

P.S. I just learned that there is a simpler version of SPOT IT! that has 6 symbols on each card and the deck consists of 31 cards (the maximum that there could be to ensure a match of one and only one symbol between any two cards!).

Here are pdf files to make the cards in the game of SPOT IT!

Spot It! cards -- Order 0 (1 card and 1 symbol, 1 symbol per card)
Spot It! cards -- Order 1 (3 cards and 3 symbols, 2 symbols per card)
Spot It! cards -- Order 2 (7 cards and 7 symbols, 3 symbols per card)
Spot It! cards -- Order 3 (13 cards and 13 symbols, 4 symbols per card)
Spot It! cards -- Order 5 (31 cards and 31 symbols, 6 symbols per card)
Spot It! cards -- Order 7 (57 cards and 57 symbols, 8 symbols per card)

Spreadsheet to print 13 Words for SPOT IT! Cards by Radigan Engineering

Richard Booth, an Englishman living near Cheltenham and a retired mathematical researcher, wrote to me in January 2021 about the Harry Potter Dobble Card Game from Asmodee. The game is exactly like Spot It! but features imagery of Harry Potter, Hermione Granger and Ron Weasley, Hedwig, wands, creatures, Hogwarts house crests and more. The company used 55 cards instead of 57, just like Spot It!
Here is Richard's method for finding the missing cards:

"First I counted the number of times each of the 57 symbols occurred (and in the process verified there were 57!). As expected, there were 15 symbols with a shortfall, with 14 of them occurring 7 times and one occurring 6 times. This last one was the unique symbol common to the two missing cards, and it was Dumbledore.

Then I looked through the cards one by one. If Dumbledore was present then it gave no information. But otherwise there had to be exactly 2 of the '14' present, so that this card could intersect each of the 2 missing cards. So I wrote down those two symbols with a line between them. The line meant they had to be on different missing cards. After looking at about 20 cards and keeping records, two sets of 7 symbols emerged with no lines between them, and these defined the missing cards. So the missing cards were deduced to be:

Dumbledore, Sorting Hat, Quill, Horntail Dragon, Hogwarts Express, Fang, Voldemort, Mirror

Dumbledore, Wand, Glasses, Cup, Professor Lupin, Hogwarts Crest, Harry’s Scar, Hagrid"