From number theory to unsolved conjectures, Kukla explores why the most powerful ideas in mathematics often emerge from simplicity, abstraction, and the courage to rethink the problem itself.
1. What’s the most mind-bending math problem you’ve ever solved, and how did it change the way you see numbers?
There is one class of numbers I find particularly fascinating: practical numbers.
A natural number n is called practical if every positive integer smaller than n can be expressed as a sum of distinct divisors of n. While this may sound abstract, consider the number 6. Its divisors are 1, 2, 3, and 6, and every number below it can be constructed from these:
1 = 1
2 = 2
3 = 3
4 = 3 + 1
5 = 3 + 2
Practical numbers share intriguing similarities with prime numbers, but my focus was on unexplored territory. I worked on an open problem: proving that every practical number is also quasi-practical—meaning the property still holds even when n itself is excluded from its divisors.
The problem resisted solution for days. Every approach failed—until I realised I needed to step back and generalise. By introducing the concept of practical sets, I was able to reframe the problem, and within that broader structure, the solution emerged.
That moment reshaped my thinking. It reinforced a fundamental truth: in mathematics, stepping back and expanding the framework often reveals what direct effort cannot.
2. You share fascinating math facts online—which one surprised you the most?
One of the most striking insights came during my early studies in logic and set theory.
We often assume that larger sets must contain more elements. Yet in mathematics, size—or cardinality—is defined through one-to-one correspondence. This leads to a counterintuitive result: the set of all natural numbers and the set of all even numbers contain exactly the same number of elements.
By pairing each natural number n with 2n, we establish a perfect correspondence between the two sets. Despite one being a subset of the other, they are equal in size.
This idea challenges intuition and reveals the depth of mathematical thinking—especially when dealing with infinity.
3. For someone intimidated by math, what’s one simple but magical idea you’d want them to experience?
One of the most accessible yet profound ideas in mathematics is the divisibility rule for 3.
At first glance, it seems like a trick: a number is divisible by 3 if the sum of its digits is divisible by 3. Many people memorise this without understanding why it works.
But the explanation reveals the beauty beneath the surface. Every number can be expressed as a sum of its digits multiplied by powers of 10. When you subtract the sum of the digits, what remains is always divisible by 9—because each power of 10 minus 1 is divisible by 9.
This insight shows that what appears to be “magic” is actually structure. Mathematics is not about memorisation—it’s about understanding the patterns that govern numbers.
4. Have you ever had a ‘Eureka!’ moment in math that felt more like art than science?
Eureka moments are central to mathematics—they are the reward for sustained, often frustrating effort.
One of my favourite examples is the so-called Eureka Theorem, proven by Carl Friedrich Gauss.
The theorem states that every natural number can be expressed as the sum of three triangular numbers. Beyond its mathematical elegance, what makes it remarkable is its history. In 1796, after discovering the proof, Gauss famously wrote in his diary: “EYPHKA! num = Δ + Δ + Δ.”
The result itself is beautiful, but so is the story behind it—a reminder that mathematics is not just logical, but deeply creative and, at times, poetic.
5. Which unsolved problem or mathematical mystery keeps you awake at night—and why?
The great unsolved problems of mathematics carry an almost mythic weight. From the Riemann Hypothesis to Goldbach’s Conjecture, they have defined entire careers—often without resolution.
Personally, I’m not drawn to chasing those monumental problems. Instead, I’m captivated by something deceptively simple: the Collatz Conjecture.
The rules are straightforward. Start with any natural number: if it’s even, divide it by 2; if it’s odd, multiply it by 3 and add 1. Repeat the process, and the conjecture claims you will always reach 1.
Despite its simplicity, it remains unsolved. Even modern advances—such as partial progress by Terence Tao—have not cracked it completely.
What makes it compelling is this contrast: a problem simple enough for a child to understand, yet profound enough to resist the most advanced mathematical tools.
Bio:
Andrzej "Mathinity" Kukla. A graduate in mathematics from the Jagiellonian University. Creator of social media profiles popularizing higher mathematics – accessible in form, yet intellectually deep. A researcher by passion, focusing on number theory and recreational mathematics.
Interview By - Irene Elina Eldhose
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