# Spot It!2020/12/25

## Rules of the Game

*Spot It!* or *Dobble* is a game where players must find the common symbol between two cards.

Interestingly, the game’s manual states that

- There are 55 cards, with each card having 8 different symbols.
- Between any 2 cards, there is exactly 1 matching symbol.

But there is no mention to how many symbols there are in the entire deck.

So how many symbols are there?

## An Upper Bound

A trivial case would be if every single card had, say, a , and each card’s other 7 symbols were all different. In this case, the deck would have total symbols.

But anyone who’s played the game knows that the deck isn’t structured like this. The answer isn’t unique, and the more interesting question is:

What is the smallest possible number of unique symbols?

is the first upper bound.

## A Lower Bound

Constructing cards one by one offers some more insight into the problem. Start with a card with 8 random symbols.

The next card constructed will contain a symbol from the first card, along with 7 new symbols.

The next card will contain a symbol from each of previous 2 cards, along with 6 new symbols.

Do the same to create another card.

This process can go on until 9 cards have been made, utilizing 36 unique symbols, and it shows that 36 is the fewest number of symbols required to build a deck of 9 cards. Unfortunately, this algorithm stops at 9 cards. It is impossible to add a 10th card without editing the some of the 9 cards already made. (Try to add a 10th card, keeping in mind that every preexisting symbol appears exactly twice.)

Thus is the first lower bound.

## Partitioning the Deck

The next idea to try involves splitting up the deck into small groups and analyzing those groups. First, one card is chosen at random.

Every other card in the deck shares exactly one of the 8 symbols with this card. For example, one group of cards all contain a , while another group of cards all contain a . This partitions the other 54 cards into 8 groups, based on which symbol is shared.

Now how do the groups interact with each other? Pick one card from the group and compare it with some cards from the group. The card must match with the cards through **different** symbols. (Otherwise the group breaks down).

This means that the size of the group can’t be more than , and there are only two ways the 54 cards can be partitioned, based on the size of the groups:

This argument assumes that there are 2 non-empty groups, so there is also the third possible partitioning , which is the trivial case already discussed.

Pick three groups of size , and call them the , , and groups. The group contains 49 unique non- symbols. There are also 49 different ways to pair up a card from the group with a card from the group. Each connection requires a unique symbol, so all 49 symbols are used to connect the and groups. These same 49 symbols must also be used to connect the and groups, and consequently the and groups. 49 symbols **must** be used to ensure the entire deck adheres to *Spot It!* rules.

Including the 8 symbols on the card originally chosen to partition the deck, **57 symbols** are used in the game of *Spot It!*.

## The *Spot It!* Geometry

There are striking similarities between the properties of *Spot It!* and the axioms of incidence geometries:

Incidence geometry | Spot It! |
---|---|

There are points and lines, and lines have points. | There are symbols and cards, and cards have symbols. |

Every line has at least two points. | Every card has at least two symbols. |

Two points determine at most one line. | Two cards cannot have two matching symbols. |

Every pair of distinct lines meet in exactly one point. | Every pair of cards has exactly one matching symbol. |

The last property, which I call the *Spot It!* axiom, heavily suggests that *Spot It!* is some kind of projective plane, although it is difficult to check if *Spot It!* satisfies some of the other axioms.

A simple projective plane can be constructed, starting with an grid of points.

Pick any point in this grid.

Pick some number of rows and columns to travel away from this point. For example, row down and columns to the right.

Continue travelling in this direction, wrapping around the grid when necessary, until it’s back at the start.

The 5 points visited determine a line, and changing the starting point and direction gives all other lines. Since is a prime number, the following are true about this geometry:

- There are points and unique lines.
- Every line has exactly points.
- For any two points, there is a unique line containing them.
- For every line and point not on , there exists a line containing that is
**parallel**to . - There are groups of parallel lines, each group containing lines.

This is an affine geometry, but can be made into a projective geometry. Consider the following group of parallel lines.

To make them all intersect, add another **point at infinity** that all these lines contain.

Every group of parallel lines has their own point at infinity, and all the points at infinity are also connected by a line.

This modified geometry has some new properties.

- There are points and unique lines.
- Every line has exactly points.
- For any two points, there is a unique line containing them.
**Every pair of lines intersects at exactly 1 point.**

Introducing points at infinity makes it so that the last property holds, and this is now a projective geometry. By reinterpreting points as symbols and lines as card, we have a structure where **every pair of cards shares exactly one common symbol**.

Setting gives a way to construct a deck of 57 cards, each card having 8 symbols, and all cards satisfying the *Spot It!* axiom. But *Spot It!* comes with only 55 cards! If a deck satisfies the *Spot It!* axiom, then every subset also satisfies the *Spot It!* axiom, and for whatever reason, the game chooses to omit two possible cards.