How to Make a Dobble Deck

What’s going on

In the card game Dobble (sold as Spot It in some countries) every round card carries eight small pictures. Put any two cards side by side and they have exactly one picture in common, never zero and never two. The first player to spot it wins the card.

Press Play above and click the symbol that is on both cards. Then ask the real question: how would you design such a deck? Scattering symbols at random fails: two random cards would often share nothing, or two.

Start small: three symbols per card

Press Fano plane: seven symbols, seven cards, three symbols on each. Draw each symbol as a point and each card as a line through its three points. One “line” is bent into a circle; all that counts is which points lie on it.

Now the rules of the game turn into rules of geometry:

• Any two cards share exactly one symbol becomes any two lines cross at exactly one point.
• Any two symbols appear together on exactly one card becomes through any two points there is exactly one line.

The second rule is the familiar one from school geometry. The first is the strange one: in this picture there are no parallel lines. A tiny world with seven points and seven lines like this is called the Fano plane, and mathematicians call such worlds finite projective planes.

Eight symbols per card: a grid with directions

Press Build a deck and choose order 7: a 7 × 7 grid of 49 symbols. The cards are straight lines on it, but the grid wraps around like an old video-game screen. A line of slope 2 climbs two squares per step to the right, and when it leaves the top it comes back at the bottom. In numbers, y = 2x + 1 means “take 2x + 1 and keep the remainder after dividing by 7”.

Every such line hits 7 grid points, and two lines with different slopes cross at exactly one. Here it matters that 7 is prime: arithmetic “mod 7” never lets two lines meet twice.

But two lines with the same slope never meet: parallel lines again. The fix: add one extra symbol for each direction, slopes 0 to 6 plus vertical. Every card gets the symbol of its direction, so parallel lines share exactly that one. Finally, one last card holds all eight direction symbols: the “line at infinity”.

Count it all up:

• symbols: 49 grid points + 8 directions = 57;
• cards: 7 slopes × 7 lines each, plus 7 vertical lines, plus 1 card at infinity = 57;
• every card: 7 grid points + 1 direction = 8 symbols.

Press Check every pair: the scene compares all 1,596 pairs of the 57 cards and finds exactly one shared symbol every time. The same recipe works for any prime n: n × n + n + 1 cards, with n + 1 symbols on each. The shop version of Dobble uses this 57-card design but puts only 55 cards in the box.

Try it yourself

1. Dobble for toddlers

Make a deck where every card has just two symbols, any two cards share exactly one, and no symbol is on every card. How many cards can it have?

Answer

Three: AB, AC and BC. A fourth card would need a symbol in common with each of them. With only two symbols on it, the only ways to do that are AB, AC or BC again (two equal cards share two symbols). A card like AD meets AB and AC but misses BC. In the language of points and lines, this is a triangle: three points, three lines.

2. Why must the grid size be prime?

Try the grid recipe with a 6 × 6 grid. Take the lines y = 2x and y = 5x, keeping remainders after dividing by 6. Where do they meet?

Answer

At x = 0, 2 and 4: the points (0, 0), (2, 4) and (4, 2). Both lines hit all three, because 2x and 5x differ by 3x, and 3x is a multiple of 6 whenever x is even. So two cards would share three symbols. With a prime like 7 this cannot happen: if (5 − 2) × x is a multiple of 7, then x itself must be.

3. The two missing cards

The box has 55 of the 57 cards. How could you work out exactly which two cards are missing, without knowing the recipe?

Answer

Count how often each symbol appears. In the full deck each one is on 8 cards. Without the two missing cards, one symbol (the one they share) is on only 6 cards, and 14 symbols are on 7. Those 14 split into two groups of 7: two symbols from the same missing card never appear together anywhere else, because every pair of symbols shares exactly one card. Each missing card is one group plus the shared symbol.

Go deeper

• The seven-point plane is named after the Italian mathematician Gino Fano (1871–1952), who studied geometries with finitely many points.
• The grid recipe works for every prime, and a cleverer version works for powers of primes such as 4, 8 and 9. For order 6 no plane exists, and for order 10 none exists either: that was settled in 1989 by a huge computer search. Nobody yet knows the answer for order 12.