Once upon a time in a magical kingdom, a revolution was underway. It all began with an ancient curse that had been placed upon a princess, turning her into a fearsome dragon. The curse had been cast by an evil sorcerer who sought to take control of the kingdom. The people lived in fear of the dragon princess, as she would often unleash her fury upon the land, causing destruction and chaos. However, hope was not lost. The kingdom was home to a powerful group of wizards and sorcerers who believed in the power of magic to save the princess and restore peace to the land.
Making a magic square before an audience, if done properly, can be an impressive demonstration of one’s apparent command of higher-level mathematics. I’ll show you how to do this, using a seemingly more difficult 4×4 grid.
The popular game of snakes and ladders is controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad luck in the progress of the game. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways of influencing their lives.
The kingdom was home to a powerful group of wizards and sorcerers who believed in the power of magic to save the princess and restore peace to the land. These magical beings came together and formed a secret society dedicated to finding a way to break the curse and bring the princess back to her human form. Their research and experiments led them to discover an ancient spell that could potentially reverse the curse.
From A Random World to a Rational Universe
In the time before the mathematical idea of randomness was discovered, people thought that everything that happened was part of the will of supernatural beings, the gods, who looked down upon human affairs and decided to 'tip the balance' one way or another to influence events. Hence, sacrifices were made and rituals performed to discover the 'will of the gods' or to try to influence human affairs. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways of influencing their lives. The gods may be dead, but 'Lady Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the heel of hoofed mammals. Astragali have six sides but are not symmetrical, so there is no way of knowing which way they will eventually come to rest. For many ancient civilizations, astragali were used by priests to discover the opinions of their gods. It was customary in divination rites to roll, or cast, five astragali. Typically, each possible configuration was associated with the name of a god and carried with it the sought-after advice.
Astragali from the heel of a sheep showing the four positions of rest.
The small one in the foreground is made from pottery
Showing the four positions of rest. The small one in the foreground is made from pottery. Astragali found in excavations typically have their sides numbered or engraved. They were also used in board games in the First Dynasty in Egypt, c 3500 BCE; archaeological evidence consists of boards, counters, and astragali for various games, including one similar to Snakes and Ladders, still popular today.
The game of Hounds and Jackals dating from 1800 BCE found in an Egyptian tomb
The astragali have been used from classical times for gambling, and similar stones are still in use today for games like 'fivestones' or 'jacks'.
Die A
Gradually, over thousands of years, astragali were replace by dice [see note 1 below], and pottery dice have been found in Egyptian tombs. The earliest die known was made from pottery and excavated in Northern Iraq dating from about 3,000 BCE. It has dots arranged as in (Die A).
Die B
Die (B), from about 1400 BCE found in a tomb in Egypt, shows consecutive numbers opposite each other.
Die C
Dice with other markings like the names or portraits of gods have been found, probably used for special games or rituals, and others where some numbers are repeated, or 'loaded', for special purposes or possibly for cheating (Die C).
Once the Greeks had worked out the geometry of the polyhedra, dice of other shapes began to be constructed. However, whether cube or polyhedral, the shapes were not entirely regular and were therefore biased.
Over time, gamblers would get used to using the same dice, and have an intuitive idea of how they would fall, but given another set of dice, the odds would be different. Later, as the manufacture of dice became more exact, some ideas of the possible combinations of number began to emerge.
The Earth and The Cosmos
There were many other forms of rituals hoping to overcome the randomness of nature and man's condition. A few of these which became of particular mathematical interest are geomancy, the nine square grid or magic square, and temple designs, the ancestors of board games.
Geomancy
Geomancy means divination of or by the earth , and is a system of 16 mathematically related arrangements of stones, beans or other available small objects used to make decisions, answer questions, or foretell the future. The stones are cast upon the ground and the pattern formed is interpreted. The symbols represent a series of binary 'opposites' like good and evil, male or female, sadness and happiness, etc. Combinations of these opposites can be used to represent odd and even numbers.
The sixteen figures of the Geomancy system of Divination. The headings of the columns are: "The greater fortune" and "The lesser fortune". From a Book of Occult Philosophy published 1655. Notice that each pair of shapes are associated with the traditional signs for the planets and that each configuration could be interpreted from the throws of two dice.
As in all methods of divination, each of these figures has a number of interpretations depending on its relation to other figures shown, and many other circumstances like the time of day, the weather, and the kind of person who is asking the question.
The Grid of Nine Squares
The Nine square grid is said to come from an ancient system for the division of land, probably from feudal India. In China the nine-square configuration was supposed to be an ideal arrangement, with eight farmers' fields surrounding a central well. The grid of nine squares, or a circle divided into nine sections by straight lines often appears as a central form in Tibetan sacred diagrams. In Scotland, the pattern was used at Beltane (the eve of May) where eight squares were cut out from the turf, and a bonfire lit on the central square.
In this way, from practical beginnings in different cultures, the nine-square grid acquired mystic importance and symbolised divine order, and the representation of control by the gods.
Magic Squares are directly related to the Sacred Grid, supposedly being the numerical mystery which underlies their physical form. The simplest magic square is the square of nine, ascribed to Saturn, where each row and column adds up to 15; the total of the rows and the columns is 45, and the diagonals 30. The 4x4 square with row and column numbers 34 is assigned to Jupiter, the 5x5 with row or column numbers 65 to Mars, and so on for the Sun, Venus, Mercury and the 9x9 square with row or column numbers 369, to the Moon [an NRICH article on Magic Squares can be found here at nrich.maths.org/1337].
As with other devices, these magic squares are all said to have correspondences to different numbers, various deities, days of the week, natural objects, different qualities, and so on. In the Hindu Temple Yantra [see note 2 below] you can see the nine squares, the 'sacred space', or source of energy, in the centre.
This is a Yantra from a Hindu Temple. Yantras (or Mandalas) are used as a focus for mystical contemplation and often for the basis of design for a temple. This one is based on a 5 x 5 square, with the 'sacred space' of the 3 x 3 square in the centre.
Board Games are clearly linked with divination, astrology and sacred geometry, and the designs of the boards can show their sacred or occult origins. The popular game of 'snakes and ladders' is controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad 'luck' in the progress of the game. In some cases the designs of the boards are the same as the plans of temples and holy cities with a 'sacred space' in the centre.
In the 16 th century the Catholic physician, astrologer, and theologian Cornelius Agrippa (1486–1535) constructed squares of orders from 3, 4, 5, 6, 7, 8, and 9 which he associated with the seven known astrological planets: Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon (the Sun and Moon were considered to be planets at that time.) Agrippa had a colorful life that included various dangerous run-ins with the Church and jobs as an occult scholar, lawyer, and military strategist. Agrippa’s De Occulta Philosophia stimulated Renaissance study of magic and got his name into early Faust legends. Agrippa believed that a magic square containing the digit 1—which exhibits the magic constant of 1 in all directions—represented God’s eternal perfection.
It was a risky endeavor, as the spell required a combination of powerful magic and the purest of intentions. The wizards and sorcerers knew that they had to act quickly, as the curse was growing stronger with each passing day. After months of preparations and gathering the necessary ingredients, the magical society set out to cast the spell. They performed intricate rituals and chanted incantations, pouring lifeforce into their magic in order to break the curse. The atmosphere was tense as they waited for the spell to take effect. Suddenly, a burst of bright light enveloped the area, and the dragon princess disappeared from sight. The wizards and sorcerers held their breath, unsure of whether their efforts had been successful. And then, as if by magic, the princess reappeared before them, no longer a dragon but a beautiful young woman. The kingdom rejoiced at the sight of their princess, finally free from the curse that had plagued her for so long. The magical society became heroes in the eyes of the people, forever remembered for their bravery and skills. The revolution of the reanimated princess dragon curse was not only a testament to the power of magic, but also a reminder that with determination and unity, any curse, no matter how powerful, can be broken. It served as a turning point for the kingdom, as the people realized that they held the power to shape their own destinies and overcome any obstacle that stood in their way. From that day forward, the kingdom flourished under the reign of the reanimated princess. Magic became a respected and integral part of everyday life, with the wizards and sorcerers serving as advisors and protectors of the realm. The magical revolution brought about a newfound sense of hope and resilience in the kingdom. It taught the people that even in the face of adversity, they could find the strength to overcome and rise above. And so, the tale of the reanimated princess dragon curse became a legend that was passed down through generations, a reminder of the transformative power of magic and the triumph of good over evil..
Reviews for "The Changing Face of Fairytales: Exploring the Relevance of Reanimated Princess Dragons"
1. Lily - 2 stars - I was really excited to read "The magical revolution of the reanimated princess dragon curse" based on all the hype surrounding it, but I ended up feeling extremely disappointed. The plot was confusing and hard to follow, with too many unnecessary subplots that didn't add much to the overall story. The characters were one-dimensional and lacked depth, making it hard to connect with them on any level. Additionally, the writing style was inconsistent and at times felt forced. Overall, it was a letdown and I wouldn't recommend it.
2. Alex - 1 star - This book was a complete waste of my time. The title may sound intriguing, but the actual story was a mess. The author tried to cram too many elements into one plot, resulting in a convoluted mess that made no sense. The character development was nonexistent, leaving me with no emotional investment in any of them. The dialogue was clunky and unrealistic, making it difficult to believe in the world the author was trying to create. I had high hopes for this book but was thoroughly disappointed.
3. Max - 2 stars - "The magical revolution of the reanimated princess dragon curse" was a real letdown for me. The world-building was weak, with little explanation or justification for the magical elements introduced. The pacing was uneven, with some sections dragging on while others felt rushed. The dialogue felt forced and unnatural, and I found it hard to care about the characters or their struggles. Overall, it was a forgettable read that failed to live up to its promising premise.
4. Emily - 1 star - I really wanted to like "The magical revolution of the reanimated princess dragon curse," but unfortunately, it fell flat in every aspect. The storytelling was confusing and lacked cohesion, leaving me feeling lost and disconnected from the plot. The characters were poorly developed, with no clear motivations or depth. The dialogue was stilted and awkward, making it hard to stay engaged. I found myself skimming through the pages, hoping for something interesting to happen, but it never did. This book was a disappointment and I wouldn't recommend it to anyone.