Category: Mathematics

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The Mitre Puzzle

A timeless geometric challenge by Sam Loyd…

The Mitre Puzzle is a classic dissection problem that asks: how can you cut a bishop’s mitre-shaped figure—a square with a triangular notch—into pieces that rearrange perfectly into a square?

Sam Loyd thought he had the answer with a four-piece solution. But there was a catch—his pieces actually formed a rectangle that just looked like a square. The illusion fooled many, but the puzzle wasn’t truly solved.

Enter Henry Dudeney, Loyd’s contemporary and fellow puzzle master. Dudeney showed a correct solution requiring five pieces, and for over a century, that was accepted as the minimal number needed.

Fast forward to 2024. Finnish mathematician Vesa Simonen shook things up by discovering several true four-piece solutions—finally cracking what was long considered impossible.

Mitre puzzle, Sam Loyd

It’s interesting how even the oldest puzzles can still surprise us when we look closer.

You can explore Vesa Timonen’s innovative four-piece solution to the “Mitre Puzzle” on his dedicated webpage.

If you want to dive into Sam Loyd’s original puzzles, you can grab his classic book here.

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Roman Numerals vs Hindu-Arabic Numbers in Psychology

What Our Brains See vs What They Read

The way our brain interprets Roman numerals and Hindu-Arabic numbers reveals an interesting distinction in psychology—especially when viewed through the lens of communication theory.

According to the psychologist and communication theorist Paul Watzlawick, signs and symbols can be divided into two categories: analogical and digital.

· Analogical signs resemble what they represent. They are intuitive and visually descriptive.

· Digital signs are symbolic. They rely on learned codes and have no visual connection to what they signify.

In this sense, Roman numerals (like I, II, III) are analogical. When we see “II”, we can immediately see two units. The visual repetition reflects the quantity directly—our brain interprets the number almost as a drawing of its value.

On the other hand, Hindu-Arabic numbers (like 2, 3, 4) are digital. The symbol “2” doesn’t visually resemble two objects—it’s abstract. Understanding it depends on prior learning and decoding. The brain treats it more like language than image.

This distinction matters. Roman numerals engage perception in a way that mimics reality. Arabic numerals, by contrast, engage abstract reasoning. The first shows, the second tells.

In daily life, we may not notice the difference—but psychologically, the visual nature of Roman numerals connects us to meaning more directly, while the efficiency of Arabic numerals supports speed, calculation, and abstraction.

In short:
Roman numerals speak to the eye.
Arabic numbers speak to the mind.

⇨ More about numbers.

roman numbers vs arabic numbers
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The Origins of Our Numerals

The Kitāb al-Fuṣūl fī al-Ḥisāb al-Hindī (كتاب الفصول في الحساب الهندي), or The Book of Chapters on Hindu Arithmetic, authored by Abū al-Ḥasan Aḥmad ibn Ibrāhīm al-Uqlīdisī in 952 CE, is the earliest known Arabic treatise detailing Indian arithmetic and the use of Hindu-Arabic numerals. A unique manuscript of this work is preserved in the Yeni Cami Library in Istanbul. The treatise also offers the earliest documentation of numerals in use in Damascus and Baghdad.

Another significant reference is found in Talqīḥ al-Afkār bi-Rusūm Ḥurūf al-Ghubār (تلقيح الأفكار برُسوم حروف الغبار), or Fertilization of Thoughts with the Help of Dust Letters, by the Berber mathematician Ibn al-Yāsamīn (ابن الياسمين), who died in 1204. In the excerpt shown below, he presents the Indian numerals, stating:​

“Know that specific forms have been chosen to represent all numbers; they are called ‘ghubār’ (dust), and they are these (first row). They may also appear like this (second row). However, among us, people use the first type of forms.”​

An intriguing anecdote about Ibn al-Yāsamīn is that he composed mathematical poems, such as the Urjūza fī al-Jabr wa al-Muqābala, to make algebra more accessible. These poetic works were not only educational tools but also reflected the rich interplay between mathematics and literature in the Islamic Golden Age.

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Aristotle’s Wheel Paradox

In the classic video we’re sharing here, you see two concentric circles drawn on a rolling wheel, all sharing the same center. As the wheel rolls along the ground, it appears that the two inner circles and the edge of the wheel cover the same linear distance in one full rotation. Strange, right? This seems counterintuitive—The inner circles have a smaller circumference, so how can they travel the same distance?

Here’s what’s really happening:

🔹 The outer wheel touches the ground and rolls without slipping. It covers a distance exactly equal to its circumference.
🔹 The inner circles don’t touch the ground. They rotate along with the wheel but don’t roll independently. Instead, they’re passively dragged along—combining rotation with slipping, not true rolling.

To help illustrate this, the diagram below replaces circles with concentric hexagons. As the outer blue hexagon rotates, it carries the smaller ones by making them slip—this slipping is shown by the dashed lines.

Aristotle wheel

A Mathematical Perspective

Mathematically, the “paradox” shows that a one-to-one correspondence between points on three distinct rotating paths doesn’t imply equal arc lengths. While each point on the smaller circles aligns with a point on the larger one, their trajectories differ due to the nature of their motion.

Conclusion

Aristotle’s Wheel Paradox isn’t a true paradox, but a reminder that intuition can mislead when dealing with motion and geometry. The apparent equal travel of the concentric circles and the wheel results not from identical rolling behavior, but from the interplay between rotation, slipping, and perception.

⇨ Further reading.

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Benford’s Law: Why 1 Comes First

Benford’s Law is a curious mathematical rule that describes how often different digits (1–9) appear as the first digit in many real-life datasets. Surprisingly, lower digits (like 1) show up much more frequently than higher ones (like 9).

Simple Formula

The probability of a digit d being the first digit is:

📌 P(d) = log₁₀(1 + 1/d)

For example, the number 1 appears as the first digit about 30% of the time, while 9 appears only about 5% of the time! This pattern shows up in finance, science, populations, and even street addresses.

A fascinating rule of nature—numbers aren’t as random as they seem!

benford's law

⇨ Further reading.

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The Symbiotic Relationship Between Physics and Mathematics

Physics created mathematics, or was it mathematics that created physics? Back in ancient times, it is evident that gravity gave rise to what we now call a 90-degree angle, a symbol of perpendicularity in two or three dimensions. This marked our hominid ancestors’ first concrete encounter with mathematics. Other rediscoveries followed, as mathematics is, at its core, a constant RE-discovery of the hidden facets of the world around us. In the natural world, physics shaped the mathematical instincts of countless builders, from bees constructing their hives to spiders weaving their webs, both sensing forces and symmetry with innate precision.
Today, however, the realms of the infinitely small and the infinitely large can only be understood through mathematics. Our senses and intuition fall short when faced with these new dimensions. Concepts such as the EPR effect, gauge invariance, and spin would remain invisible to us without the crutches provided by mathematics.

· EPR effect: A quantum phenomenon where two particles become instantaneously linked, regardless of distance.
· Gauge invariance: A principle that certain physical properties remain unchanged under specific transformations.
· Spin: A fundamental quantum property of particles that represents intrinsic angular momentum.

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The Metallic Ratios: Beyond the Golden Ratio

Many people are familiar with the Golden Ratio (φ), but it is just one member of a broader family known as the Metallic Ratios. These ratios describe a recursive relationship between the sides of a rectangle. Given a rectangle with side lengths A and B (B > A), the Metallic Ratios satisfy the equation:

A / B = (B – n × A) / A

where n is a fixed integer.

For the Golden Ratio (n = 1):
A / B = (B – A) / A
which leads to:
φ² – φ – 1 = 0
solving this gives:
φ = (1 + √5) / 2 ≈ 1.618…

For the Silver Ratio (n = 2):
A / B = (B – 2A) / A
which leads to:
ψ² – 2ψ – 1 = 0
solving this gives:
ψ = 1 + √2 ≈ 2.414…

Different values of n define other Metallic Ratios. The table below presents some of them.

Curious minds, don’t stop here—explore the fascinating properties of the other Metallic Ratios and see what patterns and surprises you can uncover!

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The Art of Puzzling: A Mind-Bending Philosophy

Puzzles have always been an essential part of human culture, not just as entertainment but as a means of sharpening the mind. From the ancient Rhind Papyrus to modern brain teasers, they challenge our perceptions, forcing us to see beyond the obvious. They teach us that even the simplest elements—dots, lines, shapes—can hold extraordinary meaning, just as they have throughout history, from astronomy to the birth of the decimal system.

At their core, puzzles embody a philosophy: they cultivate curiosity, nurture self-irony, and encourage a critical yet playful detachment from reality. Each challenge is an invitation to think differently, to question, to explore. They remind us that solutions often emerge not from brute logic but from creative leaps, from seeing patterns where others see chaos.
For those who wish to dive deeper into this world, I invite you to experiment and create your own puzzles with our online tutorials. There, you can build, solve, and enjoy the endless possibilities that come from playing with perception. After all, life itself is nothing but a grand puzzle—one we’re all trying to piece together.