The LUMS Math Circle held on November 14, 2025, examined the hidden structure behind the popular game Dobble (Spot It). The session, led by Dr. Giovanni Bazzoni and Dr. Waqas Ali Azhar, combined hands-on play with geometric thinking and counting arguments to reveal why the deck is built the way it is.

Part I — The Game and Initial Activities

Playing Dobble (Dr. Giovanni)

Dr. Giovanni began with the classic Dobble game. Students were given a deck of 57 circular cards, each featuring a selection of symbols. A group of five students distributed the cards among themselves, leaving one card at the center.

The rules were simple:

•  Each player scanned their own cards to find the exact card that matched the center card.

• When a student found a match, they shouted it out and smashed it on the card at the front; rounds were repeated several times, and the student who identified the matching card most often across rounds was declared the winner.

Observations from Activity 1

Toward the end of the exercise, Dr. Waqas asked the group what they observed. Students concluded that:

• Every card contains exactly 8 symbols.

• Any two cards share exactly one symbol in common.

These observations formed the foundation for the deeper mathematics explored next.

Small Construction Tasks

Students then moved to constructive tasks to understand these regularities more deeply:

1. Three-card mini-Dobble. Using three symbols and three cards, students tried to arrange two symbols per card so that every pair of cards would share exactly one symbol. Through experimentation they saw why two symbols per card was the only workable choice in that tiny setup.

2. Seven-card challenge. Students were given seven blank cards and asked to make a Dobble-like set without a symbol list. After some attempts and the hint to use seven symbols with three symbols on each card, one group succeeded. This exercise motivated the general counting question: How many symbols are needed for a full Dobble deck?

Part II — Geometry Behind Dobble

Symbols as Points, Cards as Lines

Dr. Giovanni then invited students to visualize Dobble geometrically:

•  Symbols = Points

•  Cards = Lines

•  If a card has a symbol, then the line passes through that point

•  If two cards have a symbol in common, the two lines intersect at a point

This led students into the world of incidence geometry, where games like Dobble can be represented as geometric diagrams.

Part III — Euclidean Geometry and Parallel Lines (Dr. Waqas)

To extend the geometric viewpoint, Dr. Waqas reminded the students of the classic Euclidean idea:

Given a line and a point not on that line, there is exactly one line through that point parallel to the original line.

But in Dobble’s geometric model, the situation is very different…

•  There are no parallel lines — every “line” intersects every other line.

•  This is a property not of Euclidean geometry, but of projective planes.

(At this point, Dr. Waqas took the class to the next conceptual stage.

Part IV — Spherical Geometry and Shortest Paths

Dr. Waqas changed gear to introduce a surprising geometric idea before returning to the Dobble counting argument.

Can a triangle have interior angles more than 180°?

Dr. Waqas asked whether a triangle can have interior angles summing to more than 180°. While impossible on a flat plane, he showed that on a sphere, it is achievable: using arcs of great circles, a spherical triangle can have a total angle sum greater than 180°. This surprised students and highlighted how geometry changes on curved surfaces.

Shortest paths on a sphere — geodesics

He then asked which is shorter between two locations on Earth: a straight line drawn on a map or a curved arc. He explained that on a sphere, the shortest path is a great-circle route, not a straight map line—this is why airplanes follow such paths. On a flat plane, straight lines are shortest, but on a sphere, geodesics are.

(After this geometric interlude, Dr. Waqas returned to the counting problem and the Dobble formula.)

Part V — Dr. Waqas Explains the Formula (N^2 + N + 1)

Returning from the spherical geometry discussion, Dr. Waqas guided the students through the counting argument that explains the total number of symbols needed for a deck with the matching property. He presented the reasoning exactly as follows:

1. Suppose each card has (N+1) symbols. Think of each card as containing (N+1) symbols.

2. If every (N+1) cards share one symbol in common, then there must be a single symbol that appears on every card — this is the special common symbol.

3. On each card, since one symbol is common to all cards, the remaining symbols are on that card number (N). Therefore, each card contributes (N) symbols that are not the universal common symbol.

4. Because there are (N+1) cards, the number of non-common symbols across all cards is
N x (N+1).

5. Adding the one special symbol that appears on every card gives the total number of symbols:
N x (N+1) + 1 = N^2 + N + 1.

Dr. Waqas emphasized the importance of that final “+1” — it is the unique symbol that appears on every card. For the Dobble deck, with (N+1 = 8) symbols per card, we have (N=7) and therefore 7^2 + 7 + 1 = 57,
which is why a full Dobble design requires 57 distinct symbols.

Key Takeaways

By the end of the session, students understood:

•  Dobble is built on deep geometric principles, not chance.

•  The spherical-geometry mini-lecture shows that familiar geometric facts depend on the surface: on a sphere, triangle angles can sum to more than 180 degrees, and shortest routes are great-circle geodesics (explaining airplane routes).

•  The counting argument (N^2 + N + 1) gives a concrete formula for the total number of symbols required when each card has (N+1) symbols and the system follows the Dobble matching rule; the “+1” is the common symbol present on every card.

Closure

The session concluded with an enthusiastic discussion, as students reflected on the surprising mathematics hidden within everyday games. They were encouraged to create their own mini-Dobble sets and explore similar patterns on their own. The event wrapped up on a cheerful note with the distribution of certificates to all participants, celebrating their curiosity and engagement.

Acknowledgments

Heartfelt thanks to Dr. Giovanni Bazzoni and Dr. Waqas Ali Azhar for guiding students from playful observation to precise mathematical reasoning. Special appreciation to Ms. Noreen Sohail, Ms. Shazia Zafar, Mr. Qamar Abbas, and Mr. Javaid Qayoom (writer of this email) for their support and organization.

Here are some highlights from the event: