Given 3 circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy: ace/bdf = 1

### Morley’s Trisector Theorem

In any triangle, the 3 points of intersection of the adjacent angle trisectors ALWAYS form an equilateral triangle (in blue), called the **Morley triangle**.

### Surprising Limit

Amazingly, this sequence of fractions converges to 0.70710678118…, or to be precise, to √2/2. The sequence is related to the Prouhet-Thue-Morse sequence.

### Prime Fractions

Did you know? You can write the number 1 as a sum of 48 different fractions, where every numerator is 1 and every denominator is a product of exactly two primes.

This problem is related to the Egyptian fractions.

### Mug to Doughnut

Showing why a doughnut and a mug are topologically equivalent…

### Very Large Numbers In Real Life

Most people know about Zimbabwe’s trillion dollar bill notes or have heard stories about Germans using worthless Marks during the Weimar Republic for wallpaper, but what few realize is that Hungary broke all the records. Just after the WWII, between 1945 and 1946, Hungary was in a state of hyperinflation, with inflation rates reaching **41.9 quintillion percent** (that is 41,900,000,000,000,000,000%). Continue reading “Very Large Numbers In Real Life”

### Math-Magic Vanishing Space

Inspired from the astrological tables, here is a new puzzle of my creation designed according to the ‘Golden Number Rules’, which is reflected in the proportion of each single piece of the game. Thanks to the balanced dimensions of its pieces, this puzzle acquires some intriguing magical properties!

This “math-magical” puzzle is composed of a tray in which the pieces are assembled.

### Bidimensional Müller-Lyer Illusion

I am working on a new two-dimensional variant of the Müller-Lyer illusion… You may be surprised to know that the *Müller-Lyer illusion* isn’t only linear: it involves plane geometry too! In fig. A shown below, the ends of the blue and red collinear segments, arranged in a radial fashion around a central point, delimit **two perfectly concentric circles**. However, for most observers, they seem instead to define a large ovoid that circumscribes another one, slightly eccentric (Fig. B). This comes from the fact that the red segments seem to stretch towards the lower part of the figure, while the blue segments seem to stretch towards the upper part of the same. As you can see, in this variant comes also into play the “neon color spreading” effect. In fact, a bluish inner oval-like shape appears within the black arrow heads (Fig. A), though the background is uniformly white.

### Sum of Infinite Power Series

Have a look at the two distinct sums of series of powers below.

Same procedure, different result accuracy levels… Can you guess what went wrong in the operation of fig. 2?

### How to ‘magically’ untie a shoelace double knot

**Topology** is a fascinating branch of mathematics that describes the properties of an object that remain unchanged under continuous “smooth” deformations. Actually, many 3D puzzles are based on topological principles and understanding some very basic principles may help you analyze whether a puzzle is possible or not.

Puzzle-Meister **G. Sarcone** created this amusing everyday-life topological puzzle to help children to easily take their shoes off.

As you know, the standard shoelace knot is designed for quick release and easily comes untied when either of the working ends is pulled. Thus, most people think that tying a shoelace into a double knot is an effective method of making the knot “permanent”. **But is it true?** Continue reading “How to ‘magically’ untie a shoelace double knot”