Project Work Best in Category 8 (Mathematics): Clique Chords

Written by: Daniel Goh Hao Yang, Lee Xuan Rong Scott, Chen Yanjun , Zhao Bozheng (3i1, 2026)

At first glance, our project seemed almost too simple. Take a circle, place an even number of points along its circumference, and connect the points in pairs with chords. What could possibly be so complicated?

As we began exploring these arrangements, however, we discovered that a simple circle could lead to surprisingly challenging mathematical questions. When the chords intersected, patterns began to emerge — and among them were structures known as cliques, collections of chords in which every chord intersects every other chord.

This became the starting point for Clique Chords, our mathematical research project. More than just investigating chords and cliques, the project gave us an opportunity to experience what mathematical research is really like: asking questions without knowing the answers, testing ideas that might not work, and gradually building our own approaches to unfamiliar problems.

From Chords to Cliques

Our investigation began by randomly connecting pairs of points around a circle. We soon realised that the intersections between chords could be represented using graph theory, a branch of mathematics that studies relationships between interconnected objects.

We represented each chord as a vertex in a graph. Whenever two chords intersected, we connected their corresponding vertices with an edge. A clique was then a collection of vertices in which every pair was connected — in our setting, a collection of chords that all intersected one another.

From this simple idea, we developed three main questions.

1）How likely was it for a randomly generated arrangement of chords to contain no clique of a particular size?

2) How would the size of the largest clique change as the number of chords increased?

3) What would happen if every chord had to intersect exactly the same number of other chords?

These questions sounded manageable when we first formulated them. Finding the answers was another matter.

When There Is No Formula to Follow

One of the biggest differences between mathematical research and the mathematics we were used to was the lack of a clear path to the answer.

In a conventional mathematical problem, we are often given the information we need and can draw on familiar formulas or methods. In our research, there was no prescribed procedure. We had to decide for ourselves what mathematical ideas might be useful, test different approaches and work out why they did — or did not — work.

We began by reading existing research papers to understand the mathematical ideas behind our investigation. This exposed us to concepts in graph theory and probability that were unfamiliar to us. Some of the papers were difficult to understand at first, especially when they introduced ideas that we had not encountered in our usual lessons.

Instead of trying to understand everything at once, we gradually worked through the arguments and identified techniques that could be adapted to our own problems. This process helped us see how mathematicians build on existing knowledge while developing new ideas of their own.

One of our greatest challenges was calculating the probability of forming particular cliques. As the number of chords increased, the number of possible arrangements grew rapidly. Simply listing and counting every possible arrangement quickly became impractical.

This was where computation became an important part of our investigation.

We developed algorithms to generate chord arrangements and identify those that met particular conditions. This was not simply a matter of getting a computer to do the calculations. We first had to translate mathematical definitions into precise instructions that a computer could follow, while also thinking about how efficiently those instructions could be carried out.

The computer could help us explore the possibilities, but it could not tell us whether a pattern was mathematically true.

That distinction became one of the most important lessons of our project. Computational results helped us spot patterns and test specific cases, but we still needed mathematical reasoning to explain why those patterns occurred and determine whether they would continue to hold in general.

What We Discovered

Our investigation eventually led to several interesting findings.

We developed a systematic method for calculating the probability that a randomly generated chord arrangement contained no clique of a specified size.

We also found that the size of the largest clique grew approximately in proportion to the square root of the number of chords. In other words, even as the number of chords increased, the largest group of mutually intersecting chords grew at a considerably slower rate.

Our final investigation considered a more restricted situation, in which every chord intersected exactly the same number of other chords. Under this condition, we established that the largest possible clique could contain at most one more chord than the specified number of intersections per chord.

These results helped us understand both the probabilistic behaviour and structural limitations of cliques in chord arrangements. More importantly, they showed us how a seemingly simple mathematical construction could lead to questions that required ideas from several areas of mathematics.

More Than Just the Mathematics

Perhaps the most valuable outcome of Clique Chords was not any single result, but what we learned about doing mathematics.

We learned that research often begins with uncertainty. There were times when an approach seemed promising, only for us to discover that it did not work as expected. Instead of seeing this as failure, we gradually learned to treat these moments as part of the research process — opportunities to rethink our assumptions and try a different approach.

We also learned to be more critical of our own findings. Seeing a pattern in computational data was exciting, but it was not enough. We had to ask whether the pattern could be explained mathematically and whether it would still hold beyond the cases we had tested.

Working as a team was another important part of the experience. As all four of us were from the same class, we could easily discuss our ideas and challenges with one another. Although we divided up different responsibilities, we continued to collaborate when we encountered difficult problems. Sometimes, explaining an idea to someone else helped us recognise a gap in our own reasoning; at other times, a discussion led us towards an approach that we had not considered before.

Ultimately, Clique Chords changed the way we think about mathematics. We began with something as simple as points and chords on a circle, but ended up exploring probability, graph theory, computation and mathematical proof.

The project showed us that mathematics is not always about finding the right formula or arriving at an answer as quickly as possible. Sometimes, it is about asking a question that does not yet have an obvious answer — and having the curiosity and persistence to find out where it leads.