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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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GHOST COLORS (2)

Take a closer look at the image below—you’re in for a mind-bending surprise! There’s absolutely NO yellow, and not even red or green in sight. (Zoom in if you don’t believe it!) The only actual colors used are blue, cyan, and magenta.
What you’re seeing is a fascinating phenomenon known as “simultaneous color contrast” and “color assimilation”.—effects that ‘trick’ the brain into perceiving colors that aren’t really there.

© Gianni A. Sarcone

When you magnify a portion of the image in Photoshop, as seen below, to the right, you see a series of black bars. Some gaps that appeared yellow at first are actually pure WHITE, and the eyedropper tool confirms that only CYAN and MAGENTA are present.

The green tint perceived in some areas is a result of the interaction between black and cyan, just as the appearance of red is due to the interplay of black and magenta. As for the yellow-looking circle, it’s actually an optical effect caused by the white space between the black bars reacting to the surrounding dark blue lines — a classic case of simultaneous contrast.

© GIanni A. Sarcone

⇨ Learn more.

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Assembly Fail: The Impossible Ikea Chair

Sometimes, following the instructions doesn’t lead to the expected result. This visual illusion explores how perception, logic, and a touch of ambiguity can turn a simple assembly into something entirely unexpected.

impossible chair
© 2002, Gianni A. Sarcone.

Now available on our Gallery shop—ideal for lovers of visual humor, design fails, and optical absurdities.

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Impossible rectangle?

Cut out the two identical, imperfect rectangles shown here—each missing two corners. Follow the lines to divide them into four geometric pieces… Then try to reassemble them into one perfect rectangle.

Sounds simple? Think again! Solve it? Tell us what made it such a brain-bender!

I’ve always had a passion for puzzles made of simple geometric pieces—especially those that seem almost impossible to solve despite the deceptively simple shapes and limited number of elements. As an Op Art artist, I find these visual enigmas a delight not only for the eye but also for the mind. For someone drawn to minimalism like me, beauty lies not just in pure form, rules, or apparent simplicity, but in the very intention of the game: to create something concrete and well-defined out of very little. And yet, at first glance, the pieces rarely seem to match the information at hand—as if something’s always missing, or as if the pieces resist aligning with your will.

Back in the ’80s, I created numerous puzzles with these paradoxical traits—some even became worldwide hits. When people would say, “Ah, so you’re the creator of that devilish puzzle?” I would always reply, “No, not a puzzle, but a piece of optical art.” Or: “No, not a puzzle, but a visual paradox.” Or sometimes: “No, not a puzzle, but a moment of zen-like reflection.”

No, I’ve never created puzzles—but rather works that turn geometry into visual meditation.

⇨ More visual enigmas to create.

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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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The Trapezoid Trap

Here’s a rather tricky puzzle—perfect for the classroom or a fun activity with your kids (and possibly an excuse to sharpen your own spatial skills).

Print and cut out the five puzzle pieces (see Fig. A), then try to fit them all into the larger shape (Fig. B) without overlapping. Yes, it’s possible… As you may have noticed, all the pieces—including the final shape—are similar right trapezoids. They do, however, vary in scale, just to keep things interesting.

Cut out the 5 puzzle pieces (right trapezoids; fig. A) in order to fit them all into the larger shape (fig. B) without overlapping.

You can download the full template in PDF format here.

The first person to post a correct solution will receive a set of our original postcard designs.

And if you find yourself strangely fascinated by these slanted quadrilaterals, you’re not alone. Even ancient Greek mathematicians toyed with shapes like these to explore proportions and symmetry. Curious about trapezoids (or wondering if they’re secretly out to get you)? Here’s a helpful read: https://en.wikipedia.org/wiki/Trapezoid

Happy puzzling—and remember, if it feels impossible, you’re probably getting close.

(Hint: Some pieces may need to be flipped over, as if seen through a mirror.)

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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.