Intriguing Geometric Dissections

Any two polygons with equal area can be dissected into a finite number of pieces to form each other. One of the most interesting dissection or geometric equidecomposition puzzles is that discovered by Harry Lindgren in 1951 (see Fig. 2 further below). As you know, in this kind of puzzles, the geometric invariant is the area, since when a polygon is cut and its pieces are distributed differently, the overall area doesn’t change.

Lindgren was the first to discover how to cut a dodecagon into a minimal number of pieces that could pave a square, when rearranged differently. His solution is very elegant, he first built a regular Euclidean pavement by cutting a dodecagon as shown in fig. 1.a, then arranging the four pieces symmetrically on the plane (fig 1.b). The tessellation achieved with these pieces corresponds, by superposition, to a regular paving of squares. The example in fig. 2 shows how the puzzle appears once finished.  Continue reading “Intriguing Geometric Dissections”

The Infinite Chocolate Bar

Although the principle of these kinds of “vanish puzzles” is really quite simple, they still confound countless numbers of puzzle enthusiasts! Related puzzles
· Math-Magic Vanishing Space

Rolling Polygons

Imagine the wheels of your bike are polygons. Then, to ride smoothly the road should be made of ‘catenaries‘ (yes, those bumpy things). It is the math professor Stan Wagon who first demonstrated this concept with a real square-wheel bike at Macalester College in St. Paul, Minnesota. Continue reading “Rolling Polygons”

“Magic” Factorials

There are many fun facts regarding the factorials. For instance:

• 0! = 1 by convention. As weird as it may sound, this is a fact that we must remember.
• The number of zeroes at the end of n! is roughly n/4.
• 70! is the smallest factorial larger than a googol.
• The sum of the reciprocals of all factorials is e.
• Factorials can be extended to fractions, negative numbers and complex numbers by the Gamma function.

It is possible to “peel” each layer off of a factorial and create a different factorial, as shown in the neat number pattern below. A prime pattern can be found when adding and subtracting factorials. Alternating adding and subtracting factorials, as shown in the picture, yields primes numbers until you get to 9! Continue reading ““Magic” Factorials”

Haruki’s Theorem

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.