## 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).

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## 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.

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## 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.

## Unlocking the Fraction

Use a trick to quickly and effortlessly determine the fraction that falls between 3/4 and 4/5.

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## Lateral Mathematics

High math skills are required to solve this puzzle…
Fill in the three boxes below using any of the following numbers: 1, 3, 5, 7, 9, 11, 13, 15.
You are allowed to repeat the numbers.

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## Illusory Geometry with Dice

Forced perspective is employed to craft a well-known object: the ‘tribar.’ Emerging from an “impossible catalog,” this object takes the form of a triangular structure, with square-section bars seamlessly joining at right angles. Constructing a tribar within three-dimensional space is an illusion; in Euclidean geometry, the sum of triangle angles always equals a flat angle.

Efforts to fashion a solid object resembling the tribar have met varying degrees of success. In this instance, our construction incorporates a deliberate ‘interruption’ that, when observed from a specific angle, creates the illusion of a complete triangle.

Consider fourteen dice. Sacrifice one by cutting to detach two faces (fig. a). Adjoin the remaining dice by gluing them together (fig. b), and affix the two faces of the truncated die onto the vertical stack of dice, as shown in fig. c.

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