The Platter of Visual Proofs
The Platter of Visual Proofs
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A Day of Visual Proofs, Mathematical Magic, and Playful Rigour
November 28 was a special day for the LUMS Math Circles community, marked by the successful organization of two separate yet thematically connected sessions—one designed for school and college-going students (Grades 8–12), and the other for undergraduate students. Across both circles, the common thread was clear: mathematics was presented not as a collection of formulas to memorize, but as a living subject driven by structure, invariance, curiosity, and joy.
The day unfolded as a celebration of ideas—some ancient, some modern, all timeless—guided by passionate instructors who invited students to see mathematics rather than merely compute it.
Part I: Grade 8-12 Math Circle
The Platter of Visual Proofs
The session welcomed students from Grades 8 to 12 into an environment that was energetic, interactive, and intellectually inviting. Led by Dr. Sarmad Abbasi and Dr. Imran Anwar (head of department), this Math Circle focused on one powerful theme: visual proofs.
From the very beginning, the session challenged a common misconception among students—that mathematics is about remembering rules. Instead, the instructors emphasized why results are true, using diagrams, cut-and-paste arguments, and geometric intuition to make abstract ideas tangible.
Discovering Patterns Through Numbers
The session opened with a deceptively simple question:
"What is the sum of the first n odd numbers?"
Students were encouraged to compute small cases:
(1)
(1 + 3)
(1 + 3 + 5)
(1 + 3 + 5 + 7)
Rather than rushing to a formula, the instructors guided students to look for patterns. Slowly, a realization emerged: the sum of the first n odd numbers forms a perfect square. This observation was not left hanging in the air—it was justified visually, using square arrangements that made the result feel inevitable rather than magical.
A similar approach was used to explore the sum of the first n natural numbers. Instead of stating the well-known formula, students were shown how rectangular and triangular arrangements of dots naturally lead to it. The emphasis throughout was not on speed, but on understanding.
Algebra, Reimagined
The session then moved into algebraic identities, many of which students had seen before but never truly understood. Expressions such as:
((a + b) ^2)
((a + b) x c)
(a x b)
were expanded not symbolically at first, but visually using area models, paper cutouts, and rearrangements. Students saw that algebra is, at its heart, a language describing geometry and structure.
A particularly engaging moment came when students were asked why certain results, often taken for granted, are “not obvious at all.” This question sparked discussion and reflection, reinforcing the idea that mathematics grows deeper when we slow down and ask why.
The Pythagorean Theorem: A Geometric Story
One of the highlights of the session was the visual proof of the Pythagorean Theorem using the classic tilted square construction. Instead of presenting the theorem as a fact to be memorized, the instructors built it step by step through geometric rearrangement.
Students watched as squares on the sides of a right triangle were cut and reassembled, revealing—almost magically—that the areas aligned perfectly. The theorem stopped being an isolated formula and became a geometric truth that students could trust.
Archimedes and the Infinite
The session concluded with a leap across centuries, introducing students to Archimedes and an infinite geometric series:
[1 + 1/4 + 1/16 + 1/64 + ...]
At first, the idea of adding infinitely many numbers felt unsettling. Wouldn’t the sum grow forever? Through a clever visual argument, students saw how each term shrinks, and how geometry can “contain” infinity.
This final discussion left many students visibly intrigued. Infinity, once an intimidating concept, became something approachable—almost friendly.
Closing and Take-Home Reflections
The session ended with thoughtfully designed homework problems that encouraged students to revisit the ideas visually at home: drawing pictures, creating charts, and explaining results to friends. The room buzzed with curiosity even as the session came to an end—a clear sign that the goal had been achieved.
Here are some highlights from the event:
Instructors:
Instructors:
Dr. Sarmad Abbasi
Dr. Sarmad Abbasi
Associate Professor, Department of Computer Science
Associate Professor, Department of Computer Science
The Magic of Hamming Code
The Magic of Hamming Code
Part II: Undergraduate Math Circle
Invariance, Information, and Mathematical Illusions
The undergraduate session took a different tone—more abstract, playful, and conceptually rich—while remaining just as engaging. This circle was led by Dr. Waqas Ali Azhar and Dr. Xavier, who explored deep mathematical ideas through card tricks, games, and coding theory.
The Baby Hummer Trick: A Lesson in Invariance
The session opened with action. Every student was given a pile of cards, shuffled them freely, and then Dr. Waqas asked them to select four cards from their pile of cards and remember the bottom card. Followed a carefully structured sequence of cuts and flips. At the end, something uncanny happened: one card always stood out, facing the opposite direction from the rest.
Instead of revealing the secret immediately, the instructors posed the key question:
Why does this always work?
Students were then introduced to the idea of assigning numerical values to states:
Face down = +1
Face up = −1
At the start, the product of all card states was +1. As the steps unfolded—cuts, flips, and turns—the product changed once and then remained invariant, no matter how chaotic the process seemed.
This was a powerful moment. What looked like a magic trick transformed into a lesson about invariance, a central idea in modern mathematics. Students saw firsthand how identifying the right quantity can bring order to apparent randomness.
The lingering question—why the bottom card always behaves differently—was intentionally left as a challenge, inviting students to revisit the logic and uncover the deeper structure on their own.
From Shuffling Cards to Bob Hummer
The discussion naturally extended to the Hummer Shuffle and the ideas associated with Bob Hummer, blending recreational mathematics with serious reasoning. The trick served as a bridge between play and proof, showing that mathematics thrives in both worlds.
The Magic of Hamming Codes
The second half of the session shifted gears into the realm of information theory, guided by Dr. Xavier. Students were introduced to the binary number system, starting from the basics and gradually building toward a profound question:
How can we detect and correct errors in transmitted information?
Using binary vectors, coloured cards, and carefully constructed tables, students learned how Hamming codes allow the detection and correction of a single error in transmitted data. The abstract idea of “distance” between code words became concrete through examples and demonstrations.
A particularly engaging activity involved students secretly choosing a number and a colour, responding honestly or dishonestly at specific moments. While a volunteer deduced both pieces of information. What felt like mind-reading was, in fact, mathematics at work.
This game uses a (7, 4) Hamming code, which encodes 4 bits of data (numbers 0-15) into 7 bits by adding 3 parity bits.
A student selects a secret number (0-15) and chooses one of the seven colours.
Volunteer shows each card one by one and asks, “Is your number on this card?”
The student must answer truthfully for six cards, but must lie for the card that matches their chosen colour.
Volunteer analyzes the “yes/no” responses (1s and 0s). Because the (7,4) humming code can detect and correct a single error, the volunteer/magician was able to identify which colour was lied about and pinpoint the original secret number.
The numbers on cards are chosen so that each number’s binary representation corresponds to which cards it appears on.
Data Cards: Four cards represent the four bits of the number (0-15).
Parity Cards: Three cards act as “check bits” for specific combinations of the other cards.
The realization that structure, redundancy, and distance can protect information was eye-opening, especially for students encountering coding theory for the first time.
A Light-Hearted but Deep Conclusion
The session closed with humour and insight, reminding students that mathematics can be both rigorous and fun:
There are only 10 types of people: those who understand binary, and those who don’t.
Laughter followed—but so did reflection.
Final Reflections
Both Math Circles on November 28 succeeded in their shared mission: to humanize mathematics. Whether through visual proofs, infinite sums, card tricks, or error-correcting codes, students were invited to think deeply, ask questions, and enjoy the process of discovery.
The sessions demonstrated that mathematics is not confined to textbooks or exams—it lives in patterns, games, stories, and structures waiting to be uncovered. Judging by the engagement, curiosity, and energy in the rooms, these circles did more than teach mathematics; they sparked mathematical imagination.
Here’s to many more such days.
Acknowledgments
Heartfelt thanks to Dr. Xavier, Dr. Sarmad Abbasi, Dr. Imran Anwar, and Dr. Waqas Ali Azhar for thoughtfully guiding students through rich mathematical experiences—ranging from visual intuition to deep structural reasoning—and for transforming playful exploration into meaningful understanding. Special appreciation is extended to Ms. Noreen Sohail, Ms. Shazia Zafar, Mr. Qamar Abbas, and Mr. Javaid Qayoom for their dedicated organizational support and for ensuring the smooth execution of both sessions.
Here are some highlights from the event:
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