In recent years, the Dobble—also known as Spot It!— has become one of the most popular board games among children and adults. Millions of copies have been sold in its various themes – in addition to the traditional one, it has versions of Harry Potter, Frozen or Star Wars. Despite its simplicity (basically whoever is the fastest wins), its design is based on an area of mathematics known as projective geometry.
Dobble has 55 cards, each with eight different symbols, arranged so that, when you pick up any two cards, they always have a single symbol in common. At the beginning of the game, each player is dealt one card and the rest are placed in a pile, face up. The first person to identify the symbol that shares their card with the one in the center pile keeps it, showing a new card in the center. The process is repeated until the cards are exhausted, and whoever has accumulated the most wins. Well, this simple hobby can be understood as a finite version of so-called projective geometry.
Projective geometry is a branch of mathematics that captures the idea of perspective, that is, how we perceive objects from our point of view as observers. For example, although the two train tracks are parallel—they always remain the same distance from each other—when you stand on top of them and look in the direction in which they are receding, you create the sensation that they are getting closer to each other. cross over the horizon.
In projective geometry this idea is formalized and it is established, as a fundamental property of space, that any pair of lines intersects at a single point. This point will be inside the space, if the lines intersect in it; or it will be a point at infinity, if they are parallel, as in the case of train tracks. Thus, the projective plane is a way of expanding the usual plane—also called Cartesian—by adding to each line a point of infinity, where said line intersects all its parallels. The union of all the points of infinity forms, in turn, a straight line at infinity—the horizon line, in the analogy of train tracks—which also has an associated extra point at infinity.
In its finite version – the one reflected in the Dobble – these geometries change a little. Lines, instead of being made up of an infinite number of points, as taught in school, only contain a finite number of them, as happens on a television screen, where any line—any image, in fact—has a number. finite number of pixels. The Dobble corresponds to a specific finite projective geometry, in which each line has exactly eight points—seven points in space plus one at infinity. In the game, points are the symbols that appear on the cards and each card is a straight line. As in the projective plane, every two lines have exactly one point in common, that is, every two cards have exactly one symbol in common.
Thus, the projective plane of the Dobble can be visualized as a plane of 7 x 7 points to which a straight line at infinity is added, with its extra point. In total, this projective plane has 7^2 + 7 + 1 = 57 points. Applying a basic theorem of projective geometry, it follows that the number of points must be equal to the number of lines; Therefore, there are also 57 lines, or in our case, 57 cards. But, Dobble is 55! Why its designers chose 55 cards instead of 57 remains a mystery. If you have curiosity, and also patience and time, you can try to find out which two cards are missing.
Javier Aramayona He is a senior scientist at the Higher Council for Scientific Research, member of ICMAT and co-director of the Mathematical Culture Unit of ICMAT.
Stefano Francaviglia He is a professor at the University of Bologna, Italy.
Timon Agate She is coordinator of the ICMAT Mathematical Culture Unit.
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