The LUMS Math Circle session on September 26 explored the fascinating world of randomness and pseudorandomness through a series of interactive activities. Led by Dr. Adam Zaman Chaudhry and Dr. Muhammad Faryad, the session highlighted how random processes appear in mathematics, computer science, and even social media algorithms.

Part I – Exploring Randomness (Dr. Adam Zaman Chaudhry)

Dr. Adam began with a thought-provoking challenge: Can we always predict the next number in a sequence? This sparked discussions about patterns, predictability, and what it means for numbers to be truly random.

Activities:

• Dice Experiments

  • To demonstrate randomness in action, students rolled dice repeatedly and recorded the outcomes.

  • First, using a single die and later throwing two dice together twenty times.

  • They analyzed their outcomes, realizing that although individual results could not be predicted, overall trends began to emerge with repetition.

• Pseudorandom Numbers

  • He then introduced the concept of pseudorandom numbers, explaining that because computers are deterministic, they cannot generate true randomness. Instead, they rely on methods such as the linear congruential formula ((ax + b) \mod m).

  • Students saw how this method creates sequences that “look” random.

• Monte Carlo Estimation of π

  • The power of randomness was then applied to estimation. Students generated random coordinates ((x, y)) in a square.

  • By checking if points lay inside a circle, they estimated π through probability by using the Monte Carlo method.

• Random Walk

  • This part concluded with a random walk. Using coin tosses, students modeled a walker moving left or right unpredictably.

  • This simple yet effective game illustrated how randomness drives uncertainty in real-world systems.

Part II – Applications of Randomness (Dr. Muhammad Faryad)

Dr. Faryad shifted the focus to real-life applications of randomness, particularly in modern technology.

Activities:

• Social Media Analogy

  • He began with an analogy to social media algorithms. Students drew slips of paper with topics written on them.

  • If the topic appealed to them, they returned it along with a duplicate slip; if it did not, they returned only the original.

  • As the activity progressed, the pile of slips began to reflect the group’s interests, mirroring how recommendation systems gradually learn user preferences.

• Approximating the Area of Irregular Shapes

  • The final activity addressed a mathematical challenge: how to approximate the area of an irregular shape.

  • Dr. Faryad explained that if an irregular shape is confined within a rectangle, random points generated within the rectangle can be used to estimate the area.

  • Using dice rolls to assign coordinates and coin flips to determine signs, students discovered that the ratio of points landing inside the irregular shape compared to those in the rectangle provided a practical approximation.

Closure

The session concluded on a celebratory note with the distribution of certificates to all participants. This closing gesture not only recognized the students’ active involvement but also encouraged them to continue their mathematical explorations with the same enthusiasm and curiosity.

Key Takeaways

• Random numbers are crucial in mathematics, probability, simulations, and computer science.

• Computers generate pseudorandom numbers to mimic true randomness.

• Monte Carlo methods allow us to approximate complex values such as π and irregular areas.

• Randomness underpins not only scientific models but also everyday technologies like AI and social media algorithms.

Acknowledgments

Heartfelt thanks go to Dr. Adam Zaman Chaudhry and Dr. Muhammad Faryad for their inspiring leadership and engaging activities that made the session both educational and memorable. Special thanks are also due to Ms. Noreen Sohail, Mr. Qamar Hussain, and Mr. Javaid Qayyum ( writer of this email) for their continued organizational support, without which such events would not be possible.

Here are some highlights from the event: