In recent years, Dobble – also known as Spot It! – developed into one of the most popular board games among children and adults. Millions of copies have been sold in the various subject areas – in addition to the traditional one, there are also versions of Harry Potter, Frozen and Star Wars. Despite its simplicity (basically whoever is fastest wins), its design is based on a branch of mathematics known as projective geometry.
Dobble consists of 55 cards, each with eight different symbols, arranged so that any two cards always have a symbol in common. At the start of the game, each player is dealt one card and the rest are placed face up in a pile. The first person to identify the symbol that shares their card with the one in the middle pile keeps it and reveals a new card in the middle. The process is repeated until the cards run out, and whoever accumulates the most cards wins. Well, this simple hobby can be understood as a finite version of the 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 perspective as observers. For example, although the two train tracks are parallel – they are always the same distance from each other – if you stand on them and look in the direction in which they are moving away, the impression is that they are getting closer to each other. Cross the horizon.
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In projective geometry, this idea is formalized and established as a fundamental property of space that every pair of lines intersects at a single point. This point lies within the space when the lines intersect within it; or it will be a point at infinity if they are parallel, as in the case of railway tracks. Thus, the projective plane is a way of extending the usual plane – also called the Cartesian plane – by adding to each line a point at infinity where the line intersects all of its parallels. The union of all the points of infinity in turn forms a straight line at infinity – the horizon line, in analogy to train tracks – which is also assigned an additional point at infinity.
In their finite version – the one reflected in the dobble – these geometries change a little. Lines are not made up of an infinite number of dots, as taught in school, but contain only a finite number of them, as is the case on a television screen, where every line – in fact, every image – has a number. finite number of pixels. The dobble corresponds to a certain finite projective geometry in which every 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, all two lines have exactly one point in common, that is, all two maps have exactly one symbol in common.
Thus, the projective plane of the Dobble can be represented as a plane of 7 x 7 points to which a straight line at infinity is added with its additional point. In total, this projective plane has 7^2 + 7 + 1 = 57 points. Applying a principle of projective geometry, it follows that the number of points must equal the number of lines; Therefore there are 57 lines, or in our case 57 cards. But Dobble is 55! Why the designers chose 55 cards instead of 57 remains a mystery. If you are curious but also have patience and time, you can try to figure out which two cards are missing.
Javier Aramayona He is a senior scientist at the Higher Council for Scientific Research at the Institute of Mathematical Sciences (ICMAT).
Stefano Francaviglia He is a professor at the University of Bologna, Italy.
Timon Agate She is the coordinator of the ICMAT Mathematical Culture Unit.
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