Magic Squares: introduction

       

             What are magic Squares ?

          Magic squares are an arrangement of numbers in a grid of various dimensions to where each row, column, and diagonal will sum up to the same number. The grid they are placed in is generally a N x N array split into N² cells.For example: a 3 x 3 square will have 9 cells meaning there will be 9 numbers in the square.

Here is an example of a 3 x 3 magic square.

     
       Top Row :                4 + 9 + 2 = 15
       Middle Row:             3 + 5 + 7 = 15
       Bottom Row:            8 + 1 + 6 = 15
       Left Column:             4 + 3 + 8 = 15
       Middle Column:        9 + 5 + 1 = 15
       Right column:            2 + 7 + 6 = 15
       First Diagonal            4 + 5 + 6 = 15
       Second Diagonal       2 + 5 + 8 = 15

       Magic squares are an exciting study of number arrangements. Mathematicians have been fascinated by magic squares for centuries.They can be both fun and educational. Magic squares have a deep relation to a branch of mathematics known as combinatorics. However,  most just use them for entertainment.                          

                A birthday trick for magic squares

All Magic squares have a few basic components that you can use to create one. First you need a starting value, that is the lowest number that will be in the square. Second you need to know the common difference between the numbers in the square.  For example in the above square it contains all of the numbers 1, 2, 3 , 4 , 5 , 6, 7 ,8 , 9  each number in the series is obtained by adding 1 to the previous.  so the common difference between terms is 1   The third component of a magic square is the dimension. For Example. The above square has 3 rows and 3 columns so its dimension is 3 x 3. 

     Now for our birthday trick we will restrict the dimension to a 3 x 3. You may also want to keep a copy of the above square so you can see the order you put the numbers into, or you can save it and print it off to use in the trick..
(there is a method for constructing magic squares without using an already created one as a reference but ill include that in my continuation of the series on magic squares)

1. Now what you will do is ask a person for the month and year they were born, or use your own birthday.You use the last 2 digits of the year they were born for simplicity(or if you are really comfortable with your mental math skills you could use the whole year)  
2. You take those 2 digits and use it as the starting value  and you use the numerical value for the month as the common difference. 
3.  And in your head you multiply the month times 4 and add it to the 2 digit year
4.  you write down the result where they cant see you writing 
5. then  you then put the piece of paper with the number on it in your pocket 
6. you then use the year and month of their birth to construct a magic square.
7. once constructed you show them that every row column and diagonal all have the same sum
8. next you have them divide that sum by 3 and tell you the answer 
9. you then pull the piece of paper out of your pocket and "magically" it is the same number

you can omit step 8 if you are comfortable multiplying  the number you wrote down in step 4 by 3 in your head

Ill show an example of this using Benjamin Franklin's birth day 01/17/1706

1.  01/17/1706
2.   .starting value =06 common difference=1
3. 4 x the month 1 + 06 = 4 + 6 = 10   (here i will multiply times 3 so i can omit step 8 since) 10 x 3 = 30
4. 30 ( not actually writing down  because this is an example)
5. also skipping this step because its an example
6. 
   

   

 7.    Top Row :                 9 + 14 + 7 = 30

       Middle Row:            8 + 10 + 12 = 30

       Bottom Row:             13 +6 + 11 =30

       Left Column:                9 +8 + 13 =30

       Middle Column:        14 + 10 + 6 =30

       Right column:           7 + 12 + 11 = 30

       First Diagonal           9+ 10 + 11 = 30

       Second Diagonal      7 +10 + 11 = 30

As you can see the magic constant is the same number we got in step 3 

Here is a side by side comparison of the square from the example and the basic square I placed at the top so you can see how you can use the basic square to construct a square for the trick. 

 

The next part in the series will be how to construct a magic square given any starting value,common difference, and dimension without using the basic square as a template. I will be posting this in about 12 hours so be sure to check back with us.