The LUMS Math Circle held on October 31, 2025, explored the fascinating theme of “Math behind Digital Communication.” The session, led by Dr. Imran Anwar (Department of Mathematics) and Dr. Naveed ul Hassan (Department of Electrical Engineering), guided participants through the evolution of communication — from ancient signal systems to modern digital encoding — demonstrating how mathematics forms the foundation of all information exchange.

Part I – The Origins of Communication

The session began with Dr. Imran engaging students in a lively discussion on what digital communication means. The participants enthusiastically shared their interpretations, ranging from telecommunication and internet data transfer to Morse code and emojis.

Activity 1 – Semaphore Game

To show how messages can be transmitted without words, Dr. Naveed introduced the semaphore antenna, an old signaling device that represents alphabets through specific arm positions.
Students used semaphore codes to:

•  Create the word “HELLO.”

•  Spell their own names.

Pairs of students then challenged each other to decode the semaphore messages they created. The facilitators explained that semaphore communication was once used in France during Napoleon’s era, where towers equipped with pulleys and moving arms relayed messages across great distances. Students also learned how coded signals were historically transmitted through pigeons and even human runners, paving the way for coded, secure communication — the precursor to modern encryption.

Part II – The Mathematics of Coding

Activity 2 – Encoding Game

In this hands-on challenge, students were divided into four groups:

•  Group A: circle and rod

•  Group B: star and crescent moon

•  Group C: triangle and square

•  Group D: 0 and 1

Each group was asked to build a complete alphabet using only its pair of symbols. They quickly realized that if each letter used only one symbol, they could represent just two letters. To encode all 26 letters, they needed longer symbol strings.

Through reasoning and teamwork, the students concluded that using five-symbol sequences was sufficient because: [2^5 = 32 > 26]
This discovery introduced them to the fundamental principle of binary encoding — representing information using only two symbols, such as 0 and 1.

Part III – Communication through Binary

Activity 3 – Digital Communication Game

Next, the concept of binary encoding was brought to life through a game. Students worked in pairs, one acting as the transmitter and the other as the receiver.

•  The transmitter selected a secret word from a given list and converted it into binary using a 5-bit code table.

•  Instead of writing, they communicated the binary message using a chosen physical method:

o Head tilts: left for 0, right for 1

o Hand gestures: thumbs up or thumbs down

o Foot movements: left or right stomp

o Facial expressions: frown for 0, smile for 1

The receiver then decoded the message by observing the gestures. This playful experiment gave students a tangible understanding of how digital devices send and decode signals — by translating physical or electrical variations into binary patterns.

Part IV – From Codes to Images

Activity 4 – Coloring Game

The final activity connected binary encoding to image representation in computers. Students were given grid sheets numbered from 1 to 100 and were instructed to fill certain squares with black color. Once the pattern was completed, they were surprised to see familiar shapes — like a cat or other images — appear from what initially seemed like random numbers.

The facilitators explained how each black (1) or white (0) square corresponded to a bit of data. This mirrored how computers store and display digital images as binary grids of pixels.

Students then explored how color is represented digitally using the RGB model:

•  Red, Green, and Blue act as the three primary color channels.

•  Each channel is represented using 8 bits, giving 24 bits per pixel.

•  This allows computers to display over 16 million colors (since (2^{24} = 16,777,216)).

They also discussed how this system extends to videos, which are simply fast sequences of these binary-coded images.

Finally, Dr. Imran connected the lesson to SETI (Search for Extraterrestrial Intelligence) — explaining how scientists send coded binary messages into space, hoping that intelligent life elsewhere might recognize mathematical patterns as a universal language.

Key Takeaways

By the end of the session, students had:

•  Learned how mathematics powers digital communication.

•  Understood binary encoding as the foundation of computers and coding systems.

•  Experienced how symbols, gestures, and colors can be represented as binary data.

•  Realized that digital images, videos, and even interstellar messages are all built on simple mathematical principles.

Closure

The event concluded with an enthusiastic discussion about how deeply mathematics shapes modern communication and technology. Students received certificates of participation, marking the close of another intellectually stimulating and memorable Math Circle session.

Acknowledgments

Heartfelt thanks to Dr. Imran Anwar and Dr. Naveed ul Hassan for their creative and insightful facilitation of this session, which beautifully connected mathematical logic to real-world technology. Their combination of engaging demonstrations and hands-on activities kept the students captivated throughout.

Special appreciation also goes to Ms. Noreen Sohail, Mr. Qamar Abbas, and Mr. Javaid Qayyum for their invaluable organizational support and coordination.

Here are some highlights from the event: