The Intrigue of Simplicity

“A world without problems is an illusion, so is a world without solutions.” – Gianni A. Sarcone

According to the second rule of Sarcone & Waeber’s puzzle-solving principles, “nothing is ever as simple as it seems.” This is why we enjoy deceptively simple puzzles that seem almost impossible to solve. Here’s a classic topological puzzle you can create and enjoy with your kids.

You can explore the full set of puzzle-solving rules by Sarcone & Waeber here: https://www.archimedes-lab.org/sarcone_rules.html

Ambigram Magic Squares

When using standard characters, the digits 0, 1, and 8 are symmetrical around the horizontal axis, while 6 and 9 are interchangeable when rotated 180 degrees.

With these digits, we can create magic squares that maintain their constant sum even when flipped, as illustrated below.

Interestingly, when these numbers are represented in LCD style, we can also include the digit 2, which resembles a 5 when inverted. This allows for the creation of magic squares with additional properties related to both 2D and 3D symmetry—whether flipped or mirrored—such as the remarkable example created by Chris Wardle.

This isn’t the first magic square to exhibit such fascinating properties; there are many variations out there. I encourage you to explore and discover your own creations online. If you have original ideas for magic squares with these unique characteristics, we would love for you to share them with us!

For those interested in learning more about the history and mathematics behind magic squares, check out this fascinating article from the Royal Institution: The Fascination of Magic Squares.

More links of interest:
[1] https://www.rigb.org/explore-science/explore/blog/fascination-magic-squares
[2] https://math.hmc.edu/funfacts/magic-squares-indeed/
[3] https://patcherymenagerie.blogspot.com/2019/07/magic-squares.html
[4] https://www.geeksforgeeks.org/magic-squares-fun-fact-and-more/
[5] https://mathcommunities.org/magic-squares/
[6] https://www.magischvierkant.com/links-eng/
[7] https://chelekmaths.com/2020/06/30/cracking-the-cryptic-joy-and-magic-squares/
[8] https://www.byrdseed.tv/magic-squares/

Magic Square for Gunners

A magic square is a grid where the sum of the numbers in each row, column, and diagonal is the same, creating a harmonious balance. A “geomagic” square, on the other hand, is a grid of geometrical shapes where each row, column, or diagonal can be assembled into an identical shape known as the “target shape”. Like numerical magic squares, all shapes in a geomagic square must be distinct.

Concept by Lee Sallows.

The postage stamp below, issued by Macau Post on October 9, 2014, pays tribute to Lee Sallows, the creator of geomagic squares.

Further reading.

Perceptual Puzzle

Size Distortion: The length and curvature of the blue curves A and B in the diagram are highly deceptive. However, the curves are congruent! This presents an intriguing variation of the Delboeuf illusion, wherein size judgment is distorted by peripheral context.

Amazing Disentanglement Puzzle

How to transform ordinary rulers into captivating feats of magic? Equip yourself with two standard 30 cm rulers, made of wood or other materials. Attach a 20 cm string to each ruler by threading it through the hole at one end (see Fig. 1). Form a cord loop around one ruler, knotting the loose end of the cord, as depicted in figures 2 and 3. Ensure the string ring is not too tight, allowing it to glide smoothly along the ruler.

Repeat with the second ruler, threading the cord through the loop of the first ruler, as shown in figure 4.

The challenge is to separate the rulers without cutting or unraveling the cords. Despite the apparent difficulties, the solution unfolds seamlessly.

This string puzzle can also be build using two plastic pipes and two curtain rings (see figure 5).


show solution

Center of Gravity (Centroid) Paradox

Suppose we shorten the base of a triangle while keeping its altitude unchanged. The center of gravity of the triangle remains the same because the gravity point is at the intersection of the medians, located 1/3 of the way above the base.

If we push this to the limit, as shown in the diagram, the triangle degenerates into a straight line… Then something strange happens; it appears as if the center of gravity jumped 1/3 of the way from the base to halfway up (as the center of gravity of a straight line is its midpoint). How is this possible?

The triangle is a two-dimensional geometric object, whereas a line is one-dimensional. Consequently, they adhere to different laws in the realm of physics (and mathematics). The successive transformations of the triangle undoubtedly converge towards a limit: the well-known segment. However, the nature of the triangle remains unchanged even after countless reductions of the base.

Is it possible to create objects out of nothing indefinitely?

Yes, but only with a geometric trick that combines perpetual motion and “magic”. All you need is a simple sheet of graph paper, which you’ll cut into three distinct pieces after going through a step-by-step procedure that allows you to create confetti indefinitely from nothing. The game can be played indefinitely in a cyclical fashion.

Continue reading “Is it possible to create objects out of nothing indefinitely?”

Aperiodic Tiling

An aperiodic monotile, humorously dubbed an ‘einstein’ (from the German term “einstein,” meaning “one stone” or “one tile”), is a single tile that covers a surface without repeating patterns. This posed a challenging question for some time: could such a tile exist, or was it impossible?

In 2023, David Smith and his team provided an answer. They discovered a simple tile called a “hat” that can achieve this aperiodic tiling. The geometric shape of the “hat” tile is based on the symmetry and edges of a hexagon, as shown in the picture. According to Smith, this tile, along with its reflection (shown in blue), enables an array of unique, non-repeating tile arrangements. The “hat” falls within the broader category of Smith–Myers–Kaplan–Goodman-Strauss tiles.