A magic square is a square arrangement of numbers in such an order that every column, row, and long diagonal of the square sums up to be the same number called the magic constant. The history of magic squares dates back to 2200 B.C. in Ancient China. There is a Chinese myth that the Emperor saw a turtle with an interesting pattern (a magic square) and called it Lo Shu, which is the name of a famous magic square. It’s likely that magic squares travelled from China to India, to Arab countries and then to Europe and Japan. Magic squares have been used in many different areas of study including philosophy, divination, natural phenonena, human behavior, and astrology.
As explained above, magic squares are a special arrangement of numbers in which every horizontal, vertical and main diagonal line sums up to be the same number, known as the magic constant, as illustrated below.
Magic squares are distinguished by their order. The order is dependent on the number of boxes that make up the magic square. For example, a magic square of order 3 is a 3 box by 3 box square. The order of the square is important when determining the magic constant. The formula for determining a magic constant is written as follows:
M (n; A, D) = .5 (n) x [2A + D (n2 - 1)]
The formula above is explained as: the magic constant for an nth order starting with integer A and entries with difference D between terms is one-half n multiplied by the sum of [2A and D x (n2 -1)]. This formula only works for general magic squares with entries in an increasing arithmetic series.
There are more ways of distinguishing the uniqueness of magic squares. One way is by designating a square as a normal magic square. A normal magic square is one that consists of consecutive numbers, starting with the number 1. Another way to distinguish magic squares is by calling the odd or even. The odd or evenness of a magic square is solely dependent on the value of n. If n is odd, then the square is odd. If n is even, the square is even.
There are a couple different techniques to build magic squares. One was discovered by a Belgian mathematician named Maurice Kraitchick. Her method, called the Siamese method, works only odd odd magic squares. It can be summed up in 3 steps:
- Place a 1 in the center of the top row.
- Incrementally put the following numbers one box above and to the right (the counting is wrapped around, meaning falling off the top returns to the bottom and falling off the right returns to the left.)
- When a square is reached that is already filled, the next number is placed directly under the current box, and the pattern is continued.
Another mathematician that developed methods for creating magic squares was an Englishman named John Horton Conway. Conway developed methods to create both even and odd magic squares.
Conway’s method for developing an odd magic square is called the Lozenge method. The directions for this method are as follows:
- A diamond is created in the central part of the square. This diamond is filled with odd numbers in sequential order, left to right, bottom to top.
- The even number that were missed are added sequentially by following the diagonals of the diamond, obtained by wrapping around the square.
Conway’s method for creating even magic squares is named the LUX method. In this method, the boxes are broken up into 2x2 squares. The numbers in the box-squares are then arranged in 3 different ways — L, U, and X. The letters represent the direction in which the sequential numbers are put.
Knowing the patterns of L, U, and X box-squares, an even magic square with an order of n = 4m + 2 where m ≥ 1(m stands for the number of box-squares).
1. Create a square array consisting of :
- n+1 rows of L’s
- 1 row of U’s
- n-1 rows of X’s
2. Exchange the middle U with the L above it
3. Start with the middle L block of the top row with the numbers 1, 2, 3, and 4.
4. Follow the Siamese method for the box squares. Arrange the numbers in each box-square in correspondence to its letter.
There are multiple different derivatives of magic squares, such as: semi-magic, pan magic, bimagic, trimagic, associative, multiplication, and addition-multiplicaiton.
Semi-magic squares are squares in which one or both of their main diagonals’ sums do not equal the magic constant, but this is the only feature that makes them not magic.
Panmagic squares are ones in which all diagonals, including those obtained by wrapping around, sum to the magic constant.
Bimagic squares are created by replacing each number (x) with its square (x2). This is not possible with every magic square because not every magic square can be taken to the second power and still have a magic constant.
Trimagic squares are similar to bimagic but instead of squaring x, x is cubed.
Multiplication magic squares are squares in which you do not use the sum of every line, but instead the product is used for the magic constant.
Addition-multiplication magic squares are ones in which a magic constant is formed for both the sum and the product of each line.
Associative magic squares are squares in which each pair of number symmetrically opposite ti the center sum to n2 + 1. An associative magic square is shown below.
Order 5 magic square.
52 + 1 = 26
20 + 6 = 26
24 + 2 = 26
Magic squares have been useful in many areas of study in ancient times, but beyond that, they have become fun puzzles to create. The information that these figures can hold is quite confusing since they are simply an arrangement of numbers. Nonetheless, magic squares have been around for many years and continues to be used in science and math today.
Bibliography:
http://mathworld.wolfram.com/MagicSquare.html
http://www.halexandria.org/dward090.htm
https://illuminations.nctm.org/Lesson.aspx?id=655
http://www-history.mcs.st-and.ac.uk/Biographies/Conway.html
https://en.wikipedia.org/wiki/Maurice_Kraitchik
Before reading this blog, I had never heard of magic squares before, but the titled pulled me in right away. The topic seemed like something I would be very interested in learning. The topic itself seems actually challenging as well, for example, I do not understand where this formula came from and seems like something I would never be able to do. It seems challenging that someone had thought about this and did all of the mathematics behind this. It is interesting that there are many methods for this one thing. Magic squares is very interesting and yet challenging and is something I would be interested in learning more about.
ReplyDeleteThe way a magic square works with all of its numbers adding up to equal the same number no matter which direction you add them is really cool. How someone was able to come up with a method of creating these squares baffles me. The methods i found most interesting were the Lozenge method and the Siamese method. Both follow a particular pattern that was easiest for me to follow but was still very confusing. The lager the magic squares become the more complex the method for solving them becomes. Magic squares are a very complicated and sophisticated form a math and Molly does a good job explaining how they work.
ReplyDeleteI have never heard about magic squares before reading this post, but they sounded interesting. The strategies to make them seem rather confusing, but I feel like if you practiced with them they could become simpler. It’s interesting how many different strategies there are to come up with the same structure. There seems to be many types of magic squares as well which is neat, but I feel like there should be some purpose to them that relates to a real life scenario. However, since there doesn’t seem to be one, I consider these truly recreational mathematics, which makes magic squares a very unique topic to read about.
ReplyDeleteBefore Molly’s post I had never heard of magic squares. After reading the post the first time, I was still a little confused on how they work. I thought they were a similar puzzle to a Sudoku puzzle, where you had to find the missing number to make the magic square the magic square. After re-reading the post though, I understood that they are simply awesome puzzles that were created by ancient Chinese people. I was wondering if the magic constant is predetermined, or if it is found out in the course of doing the puzzle. Overall I think that Molly’s post was a really cool introduction to an interesting puzzle.
ReplyDeleteUnlike a lot of people in our class, I have actually heard of Magic Squares before. My algebra teacher would usually give us some sort of bell-ringer to get the class started, and a lot of the time she chose magic squares. However, I didn’t know the history behind Magic Squares or where they originated. I also didn’t know there was a strategy for making or solving these squares. I use to plug in random numbers and hoped I could get something to work. My favorite part of this blog post was learning about the different types of Magic Squares!
ReplyDeleteMagic squares are a really interesting concept. They are very unique in how they work and seem very complicated to build. It amazes me that these complex puzzles were created such a long time ago and that the people of ancient times were able to calculate and build these magic squares. The story of how magic squares were founded does seem a little far-fetched. The pattern on a turtles shell wouldn’t of been my first guess when thinking about where magic squares originated. Nonetheless, magic squares seem to be an ancient puzzle that has been passed down over the years to become a fun and challenging puzzle for people to try to create.
ReplyDeleteMagic squares remind me a lot of Sudoku puzzles, but it’s crazy to me how you can arrange numbers one through nine to guarantee a sum across three boxes within the square that equals fifteen. Analyzing the picture provided in the post, I’ve come to the conclusion that this sum of three boxes equaling fifteen can only be possible when each of these numbers are in their specific place. However, I think you can move rows of three numbers around, just not the numbers themselves. What I mean is that if you, for example, switch the one and the seven, the whole thing is thrown off and it is no longer possible to guarantee a sum of fifteen every time. Anyways, I’m happy Molly wrote about magic squares because I love puzzles and this is a super cool puzzle!! I will definitely be solving some magic squares in the near future.
ReplyDeleteThe first thing I thought of this for this was when we were playing Sudoku in class, just because if you would add up each row and column it should be the same if you did it right. I wonder if this could have contributed to the making of Sudoku's. I have never heard of magic squares however and I liked reading the blog about them. it was confusing however reading it then looking at the pictures really lost me though. I was able to understand what they were doing, but I don't know why they were doing it. I feel if I were to take more math classes that at least one of them would mention magic squares at some point. It would be neat to see all the different variances that you could make with these.
ReplyDeleteI've never heard of magic squares before, I instantly thought of Sudoku when looking at the squares of numbers. I wasn't very good at Sudoku and this doesn't look any easier, just different. It's cool how you arrange them in a way to form a value that's the "magic constant." Making these can seem really complicated as you have to consider all sums in many directions. I also like the fact that there are different forms of the magic squares unlike Sudoku where there is only one way to play. Panmagic seems like it would easily give me a headache and frustrate me!
ReplyDeleteBefore reading this post I had heard of magic squares before. My mom, being a math teacher, used to give me small math puzzles when I was younger and magic squares happened to be one of them. Magic squares do kind of remind me of Sudoku puzzles, but at the same time are completely different. I never knew there was a trick to solving the magic puzzles. When I was younger I always plugged in random numbers and hoped for the best, if my first combination didn't work I would try to rearrange a bit and see if something else did. However when I did magic squares as a kid I did 3x3 or maybe 4x4 never much harder than that.
ReplyDelete