The card game Dobble – Designed in 2009 by Denis Blanchot and Guillaume Gille-Naves-is based on speed, observation And reflections, they can play from 2 to 8 players and exist 5 modes main. The deck is composed of 55 cards round, each with 8 symbolsand any pair of cards random shares always a only symbol. The goal is to quickly identify the common symbol and take, discard or place the paper according to the rules of the chosen mini-game. But how is it possible that every pair of cards always have a common symbol? The answer lies in the mathematical structure at the base: the so -called finished projective plans. In this article we see how to play Dobble, as you can create a deck of cards in which each pair of cards has a single symbol in common and because to do this we need exactly 57 different symbols.
How to play Dobble: the 5 game modes
Dobble is a game of cards from 2 to 8 players and has 5 official ways To play: “The infernal tower”, “The well”, “The boiling potato”, “Take them all!” And “the poisoned gift”. In every mini-game, the challenge is always the same: individuate The Symbol in common between two cards plus quickly of others, make him aloud and then do aaction. Which? It depends on the chosen mini-game.
In “The infernal tower“, For example, the common symbol must be identified between the paper you have in front of him and the central deckvisible to everyone. If you find it before the others, yes takes The central deck card and puts it on your pile. The goal, here, is to accumulate the greater number of cards. In the mini-game “Il Pozzo”, on the other hand, the players point to discard your cards until all ended up. In this game, at each turn, you turn the first card of your deck and you are looking for the common symbol with a card located in the center of the table. Those who find it first declares the symbol in common and then square your card above the one in the center. But how is the deck built to ensure that every pair of cards always share a symbol?
The smallest as possible deck as possible: three cards with two symbols
To understand the logic of Dobble decks, let’s start with a reduced version of it, that is, consider the simplest possible deck: three cards, each with Two of the three possible symbols. To build it, we create a card containing two symbols, for example a sun and lightning. For the second card, we want me to share a symbol with the first, so let’s put the sun, but to make it different we insert another symbol, for example a moon. The third card must share a single symbol with each of the previous ones, therefore he will contain the lightning (in common with the first) and the moon (in common with the second).
In total, we have three cards And Three distinct symbols. We can therefore represent our deck as a trianglewhere i symbols are the leaders and the cards are the sides. In this case the paper with the moon and the sun will correspond to the blue side of the triangle, the one with a moon and lightning will be the green side and finally the one with lightning and a sun the red side. Each pair of card shares only one symboljust like each side of the triangle has a single summit in common with others. The blue side and the red side, for example, have only one point in common – the summit with the sun – just like the blue paper and the red paper have in common only the symbol of the sun. Now let’s go to complicate the situation a little. If instead of two symbols on each card we put three?
If we put three symbols for paper, we need seven cards
If we consider a deck in which there are 3 symbols on each card, we could represent our deck again as a trianglebut adding from the lines In addition.
Also in this case, each paper correspond to one line Colored and each card has only one element with the others. The red paper (sun, lightning, rainbow), for example, has in common only the rainbow with the yellow paper (rainbow, heart, round) and this applies to each pair of cards, or lines. Looking at the figure we can see that, to have 3 symbols for each cardwe need to create a deck with 7 cards (there are 7 different colored lines) and 7 different symbols In total.
The structure that allows us to represent the deck in this way, through a diagram, is called Plan of Fanofrom Italian mathematician Gino Fano who studied it. It is an example of structures called finished projective plans And taking advantage of the properties of these structures we can create deck decks with how many symbols we want for each card. For a deck with 8 symbols For each card, like that official of Dobble, we will need to 57 different symbols with each other.
To build the official Dobble deck you need 57 different symbols
As we said, a Dobble deck It can be represented with a projective plan. If we wanted to represent the official deck with its plane, we would get the graphic designer, much more complex than the Fano plan that we saw before.
But how did we get there? Everything is fine projective plan (and therefore each deck) is defined by a number called order. The order corresponds to the number of symbols on each card, to which it goes Subtracted 1. In the case of Plan of Fanowe had 3 symbols on each card, then order era 2 (= 3 – 1). In the Standard deck There are 8 symbols on each card, then theorder of the projective plan is 7 (= 8 – 1). In fact, the one in the graph is precisely a projective plan of order 7.
To find out how many cards we will have to print in total and how many different symbols we will have to use, we can put ourselves counting one by one lines in the graphics or we can use a property of the projective plans that tells us that the Total number of lines and points On the plane it is given by the formula: Order + order + 1. So, in a deck, the Total number of cards (the lines) and the Total number of symbols (points) can be obtained by applying this formula.
In the Fano plan, this means that there you need 7 (= 2²+2+1) cards And 7 different symbolsjust as we saw before. In the case of Standard deck of Dobble, we will need 7² + 7 + 1 = 49 + 7 + 1 = 57 different symbols between them and as many cards. But … the Dobble cards are only 55! How is it possible?
The reason why they “miss” two is a mysteryhas not been clarified by the producers and could be linked to Typographical reasons. But it is not a problem, the absence of two cards does not question the mathematical theory: by removing some cards from the deck, it does not stop the rule of a symbol in common for each pair of cards, but it simply means that for each deck there will be gods Symbols a little more uncomfortable to findbecause they will be present in 7 cards instead of 8.
We try to understand it by taking as an example the case of the three symbols and seven cards.
If we consider the Plan of Fano And let’s delete, for example, the green sideIt is as if we had eliminated the paper with the paper with the moon, the heart and the lightning. We can see from the diagram that, despite this, no symbol has remained “isolated”, but is touched by at least two colors. This means that each Continuous symbol to appear in one couple of cards and the game continues to hold up. What we can see is that the moon, heart and lightning are touched by only two colors, that is, unlike the others symbols, they appear in two cards and not on three, for this they will be rarer!
