How to create a two-digit magic square from non-repeating numbers without zeros

This article explains the method for creating magic squares from two-digit numbers.

Using this method increases concentration and promotes logical thinking in both children and adults. You don't have to be a mathematician to learn how to create a two-digit magic square. Read the previous article "The Magic Square Luo Shu Reveals its Secrets" to find out more about the requirements for creating magic squares.

A magic square is formed from the base table. The base table is formed from the first column. The first column consists of three numbers with different digits. The vertical rows of digits in the columns must show the keys.

So let's create a magic square using just two main keys: 7-1-4 and 2-5-8.

We derive the third column from the second column using the same procedure and get the base table:

Such a systematic procedure for forming a base table gives us:

• the same sum in the horizontal and vertical rows
• nine numbers whose sets of digits are not repeated.

Conversion of the base table into a magic square

First of all, you have to decide which number of the base table will be in the middle of the magic square. This number is determined by the following steps:

Step 1. Add each lower number in the base table to two other numbers in the table. All three addends must be in different horizontal and vertical rows. Each lower number gives two combinations of such an addition.

Step 2. Among these six received combinations pay attention to two in which the sums of the numbers are the same. These combinations add up to 135. Among the six addends of these two combinations, you can find two identical numbers. In this case, it´s the number 45. So, this is the central number of the magic square. The absence of two identical numbers means that the base table doesn’t have the property of being converted into a magic square.

Step 3. Now it isn´t difficult to form a magic square. To do this, you have to rearrange the horizontal and vertical rows of the table so that the number 45 is in the middle. There are in each case three variants for rearranging the rows and columns.

Let´s go back to the base table. Each complete section consists of three subsections. To get the second section, it's enough to increase each digit of the numbers in the first section by one. In the same way we get the third subsection from the second subsection. As a result, we get a complete section of three subsections. Each of the three horizontal rows of numbers adds up to the sum 495. (You can similarly create two more subsections from the finished magic square). Sometimes you can directly get the finished magic square of the next subsection by increasing all digits of a subsection by one.

Complete section of three subsections

The digital values of the numbers of the subsections

A recommended video: 

Creation of Magic Squares

Exercise: 

1. Convert the second and the third subsection into a magic square.
2. Create your own magic square using different combinations of the following keys: 1-4-7, 2-5-8, 3-6-9.