A magic square of squares is a mathematical arrangement of numbers in a square grid, where the sum of the numbers in each row, each column, and both diagonals is the same. However, in a magic square of squares, instead of using normal numbers, each cell in the grid contains a perfect square. The concept of magic squares dates back centuries and has fascinated mathematicians and puzzle enthusiasts alike. While the magic square of squares is a more specific variation, it adds an additional layer of complexity and intrigue to the puzzle. The origins of the magic square of squares can be traced back to ancient times, with examples found in different cultures and civilizations around the world. These squares have been used in various contexts, including mathematics, art, and even divination.
This Halloween, my three-year-old daughter and I are both dressing up as witches. In doing so, I hope to deepen her sense of opportunity and possibility in the world that lies before her.
Protestant authorities dismissed this ritual period as Catholic superstition, and some states re-designated 31 October as Reformation Day in commemoration of Martin Luther s momentous initial challenge to the Papacy. presidential campaign, Hillary Clinton was often either satirically depicted as a witch or was outright accused of committing acts, such as child murder, that have been associated with witches for centuries.
These squares have been used in various contexts, including mathematics, art, and even divination. The construction of a magic square of squares is a challenging task. It requires sheer mathematical skill and an understanding of number properties.
History Of The Witch: Her Story
Halloween can be traced back to the celtic festival of SAMHAIN, which marked the end of harvest and to prepare for the hard winter ahead. There are many traditions around this time of year, some that go back hundreds of years. A part of Halloween season is dressing up, and witches seem to go hand in hand with halloween - but why? I wanted to look at where the term “witch” came from and what is the history of this feared being.
One of the first places we see witches come into recorded history is the Bible (931 B.C. and 721 B.C.) The story is of King Saul who looked for the Witch of Endor to summon the dead prophet Samuel’s spirit and to help him conquer the Philistine army. The witch did show up, and did indeed rouse Samuel’s spirit, but she foretold the death of Saul and his sons. And the next day, Saul’s sons died in battle, and Saul committed suicide.
In other Old Testament verses, it cautions and warns against witches “thou shalt not suffer a witch to live.”
These stories may have likely made worshippers question and fear witches, seeing that witchcraft was evil and associated with the Devil.
In the 400s, Saint Augustine of Hippo, who was influential in the church, argued that only God had the divine power and witches or Satan did not have the power to match. The church had decided that they need not fear witches or bother investigating witchcraft further.
Then came along Thomas Aquinas in the 1200’s , who believed magic existed. He spoke of Satan and his demons could tempt people into harmful magic, using sex. Upon reading some of this history, it’s frankly disturbing how fascinated the church was with sex and seeing it as dirty - better blame the women then! This act was linked to witches.
Later, we see that the people in power who prosecuted witches, had often quoted Aquinas more than any other source.
This delightful rulebook (please note my sarcasm here!) was written by Catholic clergyman Heinrich Kramer and first published in 1486. Malleus Maleficarum - translation - “Hammer of Witches” agrees with Aquinas’s beliefs and its purpose was to gather up Satan’s tempted humans, who were used sexually and now they are all demon-y and yucky and must be destroyed….or something to that effect!
This book argued that the weaker gender a.k.a women - were more likely to succumb to the temptations of the flesh and demons far easier than men.
After the publication of this rulebook, about three quarters of the people prosecuted as witches were women.
Now came the witch hunt - everyone was scared/intrigued about witches, that during the 1400s the pursuit of witches was in full force. Across Europe this spread, using the book as a reference - women were accused, under torture and then executed. We see from accounts that in particular the women who were targeted were either opinionated, single women, widows, healers, and midwives too!
Between the years 1500 and 1660, up to 80,000 suspected witches were put to death in Europe. Around 80 percent of them were women thought to be in cahoots with the Devil and filled with lust. In Scotland, from 1500s to the 1700s, the number of accused witches reached four to five times the European average. Shocking! I used to live in a coastal town on the east coast called Pittenweem, which aside from its beauty is known as a place that trialed witches. Also if anyone has been to Edinburgh, we see their blood thirsty history does have witch trials also. I think in Scotland we are a god-fearing people, or at least we were a god-fearing people! So even though the idea of witches started off from the Catholic Church, during this time of the high witch trials - Scotland was moving to Protestantism. As people were urged to convert to the faith, witch hunts were encouraged by King James who ruled at the time, stating that he himself was a subject of witchcraft when he came back from his wedding voyage.
The burning of three witches in Baden, illustrated by Swiss clergyman Johann Jacob Wick in 1585.
(Wikimedia Commons)
One of the key elements in creating such a square is ensuring that each number in the grid is a perfect square. As with traditional magic squares, the magic square of squares follows certain rules. The sum of each row, column, and diagonal should be the same. This sum is often referred to as the magic constant. The challenge lies in finding the perfect combination of squares that fulfills these criteria. The magic square of squares has captured the imagination of mathematicians who continue to explore its properties and create new variations. The quest to find larger magic squares of squares has become a pursuit that showcases the creativity and brilliance of these mathematical minds. In conclusion, the magic square of squares is a fascinating mathematical concept that combines the art of puzzle-solving with the properties of perfect squares. It represents a complex and challenging puzzle that has intrigued mathematicians for centuries. Exploring the properties and creating larger magic squares of squares continues to be a topic of research and fascination in the mathematical world..
Reviews for "Creating Magic Squares: Algorithms and Techniques"
1. Jane - 1 star
I found "Magic square of squares" to be extremely confusing and difficult to follow. The explanations were convoluted, and the author assumed a level of mathematical knowledge that I simply do not possess. The book was filled with complex formulas and equations that were not adequately explained, leaving me feeling frustrated and overwhelmed. I would not recommend this book to anyone who does not have a strong background in mathematics.
2. David - 2 stars
While "Magic square of squares" may be intriguing for someone with a deep passion for mathematics, I found it to be lacking in practicality and relevance to everyday life. The author delves into complex mathematical theories without providing any real-life applications or examples. Additionally, the book lacked clear organization, making it difficult to navigate and understand. I was hoping for a more accessible and user-friendly approach to the topic, but unfortunately, I was left disappointed.
3. Emily - 1 star
I have to admit, I was excited to delve into "Magic square of squares" as it promised to explore the fascinating world of magic squares. However, the book was a major letdown. The author seemed more interested in showing off their knowledge of intricate mathematical proofs than in engaging with the reader. The explanations were overly technical and lacked any sort of visual aids or illustrations, making it incredibly difficult to grasp the concepts being discussed. Overall, I would not recommend this book to anyone looking for a beginner-friendly introduction to magic squares.