Dobble: a game played by kids, but still very popular among adults. In the game, you have to draw two random cards and place them face-up on the table between all the players. Then, you have to look for the identical symbol between the two cards. Between every two cards, there is always only one identical symbol. However, why do the Dobble cards always match? And how come there is only one identical symbol? Mathematics gives us the answer!
What is Dobble?
For those of you who have never played the game Dobble, a small explanation of the game will follow here. Dobble is a card game that consists of 55 cards. Each card contains eight different symbols out of fifty available symbols. An example of a card can be found in the picture at the beginning of the article. Moreover, there is only one identical symbol the same between two certain cards and it is up to you to find out which one. In the picture at the beginning of the article, the ‘zebra’ is the same on both cards. In the game, two random cards are drawn and the first person to find the identical symbol between the two cards gets to keep the cards. The player that has the most cards in the end, can call themself the winner of the game.
Earlier adaptations
The earliest adaptation of the game goes back to 1850. In this year, Reverend Thomas Penyngton Kirkman proposed the following problem in The Lady’s and Gentleman’s Diary: ‘Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.’. The combinatorial problem is now referred to as Kirkman’s schoolgirl problem and was solved by Kirkman himself a few years before.
Another adaptation was made in 1976, when the French math enthusiast Jacques Cottereau made the Game of insects. It had a deck of 31 cards with six images of insects in each and every two cards had exactly one insect in common. Later, in 2008, the game Dobble was created as an adaption of the Game of insects.
Both of these games are an example of how one can combine $n$ symbols in groups of $p$ symbols, such that no combination of $q$ symbols that appear in one combination can be repeated in another.
Geometry & Dobble
Now, how can it be that in Dobble only one symbol matches on each card? The answer lies in Geometry! In order to interpret the set-up of the game in a mathematical language, point-line geometry is useful, which is restricted to finite geometries (i.e. geometries with a finite set of points). A point-line geometry consists of a set $S = \{p_1, \dots, p_n\}$, whose elements are called points, and a family of subsets $\{L_1, \dots, L_m\}$ called lines. These should satisfy the following axioms:
- A1: Two points are in at most one line;
- A2: Two lines meet in at most one point;
- A3: A line contains at least 2 points;
- A4: There are 4 points such that no 3 of them are on a line.
If we combine this point-line geometry with Dobble, we see that the following things correspond. In geometry, there are points and lines and in Dobble there are symbols and cards, respectively. Moreover, we see that in geometry any two lines meet in one point, while in Dobble any two cards have one symbol in common. Finally, in geometry, any two points determine a line and in Dobble any two symbols belong to exactly one card. What we see here, is that we thus have the strong property that any two lines meet. For this to happen, we need the projective plane over $\mathbb{F}_p$.
The projective plane over $\mathbb{F}_p$
You probably know the Euclidean plane, which is a flat two dimensional space that extends without limit in all directions. In this Euclidean plane, for any pair of distinct lines in the plane, they either have a single point of intersection or they are parallel. However, we want any two lines to meet. This is found in a Projective plane, which uses the idea that parallel lines meet at infinity. So, for any pair of distinct lines in the plane, there is a single point of intersection. Next, the points, lines, and points in a line are counted in the Projective plane over $\mathbb{F}_p$. This gives the following propositions:
- Proposition 1: The projective plane over $\mathbb{F}_p$ has $p^2+ p + 1$ points and $p^2 + p + 1$ lines.
- Proposition 2: Each line has $p + 1$ points and each point is contained in $p + 1$ lines.
Mini-Dobble
To start, a small version of Dobble gives you the idea of the Mathematics. This small game is built using seven different symbols which are distributed among seven different cards, with three symbols per card, such that any two given cards share precisely one symbol. As each card has three symbols, this corresponds to each line having three points. Proposition 2 thus implies that $p=2$. This plane of order 2 is also called the Fano plane. Following the propositions, there are indeed three points and seven lines. In the diagram below, we can see how each line in the plane has three points with exactly one matching point with any other line. It is seen here that in this smaller version of Dobble, the formulas work.
Build Dobble using geometry
So, now let’s build Dobble using geometry. In Dobble, each card has $8 = 7 + 1$ symbols, which means that $p = 7$. This gives us that the whole deck has $7^2 + 7 + 1 = 57$ symbols. The Projective plane looks like this:
https://demonstrations.wolfram.com/ProjectivePlanesOfLowOrder/Why does Dobble only have 55 cards?
Hence, Dobble is built using a Projective plane. However, the game could apparently have 57 cards, but only has 55 cards. The reason for this is not quite known. Some believe that this is due to fabrication constraints, as a standard deck of cards has 52 cards, two jokers, and one advertising card, making a total of 55. But in the end, it doesn’t matter that much, as the game can also be played if we remove cards from the deck. The important thing is that we enjoy playing the game!
Sources
https://www.uu.nl/sites/default/files/Cecilia%20Salgado%20HandoutNWD2022.pdf
https://mickydore.medium.com/dobble-theory-and-implementation-ff21ddbb5318
http://thewessens.net/ClassroomApps/Main/finitegeometry.html