Diophantine Equations: The Pursuit Of Integer Solutions

The May 16 session of the LUMS Math Circle introduced students to the captivating world of Diophantine Equations—mathematical puzzles that ask for integer solutions. Led by Dr. Shaheen Nazir and Dr. Waqas Ali Azhar, the session unfolded as a mathematical adventure where students followed a trail of logic through puzzles, modular patterns, and linear equations to uncover hidden answers.

The theme— “The Pursuit of Integer Solutions”—was not only a mathematical goal but also the thread that tied each activity together, guiding students from one problem to the next in a narrative of curiosity and discovery.


Key Themes and Concepts:


Highlighted Problems and Activities:

Rather than isolated exercises, the session’s activities formed a connected journey, starting from simple observations of modular patterns and building up to rich, layered problems involving the Chinese Remainder Theorem and the Euclidean Algorithm.


The Beginning: Shifted Tables and Remainders

The adventure began with an Activity, where students constructed a shifted multiplication table of 7. By adding numbers 0 through 6 to each multiple of 7, a beautiful pattern emerged—every column repeating a cycle of remainders when divided by 7. This hands-on exercise introduced students to modular arithmetic, not through definitions, but through discovery.

Students quickly noticed:
 "Every number in Column 1 is a multiple of 7, with a remainder of 0."
 "In Column 2, every number gives a remainder of 1 when divided by 7."

This visual and intuitive activity laid the groundwork for formal congruence notation, such as:
 a ≡ b (mod n)


Spotting the Hidden Patterns

In Activity 2, a sequence 2, 11, 20, 29, … challenged students to think differently. They had to recognize it as a shifted table of 9, shifted up by 2. This short exercise revealed how arithmetic sequences can be recast as shifted tables, tying back to the modular thinking from the first Activity.


Modular Mysteries: Can You Find Me?

Next came two delightful riddles:

Activity 3:

“I am the smallest number such that... when divided by 3 leaves remainder 1, by 4 leaves 2, by 5 leaves 3, and by 6 leaves 4.”

Activity 4:

“I am divisible by 2 but give a remainder of 1 when divided by 4.”

These problems were not just fun; they introduced students to systems of modular equations and gently led them toward the Chinese Remainder Theorem—without ever naming it. Students built tables, looked for overlaps, and celebrated when they found the elusive number that satisfied all the constraints.

These modular puzzle sets set the tone for the rest of the session: search for patterns, build equations, and explore.

 

Pirates, Fights, and Dividing Gold: A Modular Tale

Activity 5 brought storytelling into math. Ten pirates try to divide gold coins:

This dramatic progression wasn’t just for fun. Students modeled it with three congruences:
 x ≡ 4 (mod 10)
 x ≡ 1 (mod 9)
 x ≡ 0 (mod 8)

They used shifted tables from earlier to track possibilities and intersect sets. The pirate problem connected beautifully to the earlier mystery number puzzles, demonstrating how real-world stories can be reframed as systems of modular equations.

Lights, Patterns, and Time: Synchronizing Events

Activity 6 transitioned from pirates to blinking lights. Students imagined bulbs turning on at regular intervals and asked:

“After how long do all the lights blink together?”

This was a subtle shift from congruence to least common multiples—yet still connected to previous modular work. It showed how modular reasoning helps with time-based problems, schedules, and synchronization.


Jumping Mario and the Linear Diophantine Equation

Activity 7 transformed the abstract into the playful. Mario can jump 8 or 3 meters, and a Goomba stands 10 meters away. The question:

“Can Mario reach the Goomba?”

This prompted the equation: 8x + 3y = 10
 Now the story moved from modular arithmetic into the world of Linear Diophantine Equations.

Using the Euclidean Algorithm, students tested solutions, guessed values, and confirmed that yes, Mario can reach his target. The activity demonstrated how linear equations with integers model movement and choice.

Squidward’s Shoe Store: Another LDE in Disguise

Activity 8 asked:

“Can Squidward buy and sell shoes in bundles of 12 and 30, and still end up with exactly 4 left?”

This transformed into an LDE: 12x + 30y = total + 4
 Students experimented with different values, applying logic from Mario’s problem and looking for integer solutions using the GCD.


Mastering the Method: Euclidean Algorithm and Beyond

In Activities 9 and 10, students formally learned the Euclidean Algorithm—finding GCD(5079, 4020)—and used it to solve 2x + 7y = 50. After finding a single solution, they explored how to generate all solutions using parameterized expressions. For many, this was their first glimpse into infinitely many answers to a single equation—an eye-opening moment.

Take-Home Problems:

To deepen the learning, students were given a set of exploratory problems, including:

These challenges encouraged students to extend their classroom experience into independent problem-solving.


Conclusion:

This session was more than just a lesson on number theory—it was a story of how problems unfold, connect, and reveal hidden patterns. From tables and remainders to pirates and Mario, each activity built naturally on the previous one, guiding students from curiosity to mastery.

Through storytelling, problem-solving, and mathematical exploration, this session exemplified the spirit of the LUMS Math Circle—where mathematics is not only learned but lived.

 

We extend our heartfelt gratitude to Dr. Shaheen Nazir and Dr. Waqas Ali Azhar for leading this enriching session with such clarity, creativity, and passion. Their ability to turn complex mathematical ideas into accessible and engaging explorations truly inspired our participants. We also sincerely appreciate the unwavering organizational support of Miss Noreen Sohail and Mr. Qamar Abbas, who behind-the-scenes efforts ensured the session ran smoothly.

A special note of thanks goes to Mr. M. Javaid Qayoom, whose thoughtful and detailed recap has beautifully captured the spirit and substance of the session, preserving it as a lasting reflection of the day’s mathematical journey.

Here are some highlights from the event: