Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).

Question

Homi · Accepted Answer

We can make a magic square for any nine consecutive numbers. We will adjust a basic magic square to do this.

Step 1 — Understand a Magic Square

A magic square is a grid of numbers. The numbers in each row add up to the same total. The numbers in each column also add up to this total. Numbers along the two main diagonals also add up to this total. This special total is called the "magic sum".

Let us start with a simple magic square. We will use the numbers from 1 to 9. The middle number in this set is 5. The magic sum for this square will be 15. Here is the classic 1-9 magic square:

Diagram 1

Let us check one row, one column, and one diagonal. For the top row: $8 + 1 + 6$ $= 9 + 6$ $= \mathbf{15}$ For the middle column: $1 + 5 + 9$ $= 6 + 9$ $= \mathbf{15}$ For the main diagonal from top-left to bottom-right: $8 + 5 + 2$ $= 13 + 2$ $= \mathbf{15}$ All sums are 15. This is our basic magic square.

Step 2 — Find the Middle Number of Your Set

You will be given any set of nine consecutive numbers. Consecutive means they follow each other. For example, let us take the numbers from 3 to 11. These numbers are 3, 4, 5, 6, 7, 8, 9, 10, 11. The middle number in this list is the fifth number. Let us count: 3 (1st), 4 (2nd), 5 (3rd), 6 (4th), 7 (5th). So, the middle number for the set 3-11 is 7.

Let us find the middle number for another set, 9-17. The numbers are 9, 10, 11, 12, 13, 14, 15, 16, 17. The middle number is the fifth number. Counting from 9, the fifth number is 13.

Step 3 — Calculate the Difference

We have our basic magic square using numbers 1 to 9. Its middle number is 5. Now, we have a new set of nine consecutive numbers. Let its middle number be $M$ . We need to find how much bigger or smaller $M$ is compared to 5. We calculate the difference, let us call it $d$ . $d = M - 5$

For the set 3-11, the middle number $M$ is 7. $d = 7 - 5$

$\boxed{d = \mathbf{2}}$

For the set 9-17, the middle number $M$ is 13. $d = 13 - 5$

$\boxed{d = \mathbf{8}}$

Step 4 — Build Your New Magic Square

Take the basic 1-9 magic square from Step 1. Add the difference $d$ (calculated in Step 3) to every number in this basic square. This will create your new magic square. The middle number of your new set will automatically be in the center of the square. The new magic sum will be $15 + (3 \times d)$ .

Let us use the set 3-11. The difference $d$ is 2. Our basic 1-9 magic square is: 8 1 6 3 5 7 4 9 2

Now, we add 2 to each number: $8+2 \quad 1+2 \quad 6+2$ $3+2 \quad 5+2 \quad 7+2$ $4+2 \quad 9+2 \quad 2+2$

This gives us the new magic square: $10 \quad 3 \quad 8$ $5 \quad 7 \quad 9$ $6 \quad 11 \quad 4$

Let us check its magic sum. The new magic sum should be $15 + (3 \times d)$ . $15 + (3 \times 2)$ $= 15 + 6$ $= \mathbf{21}$ Let us check one row of our new square: $10 + 3 + 8$ $= 13 + 8$ $= \mathbf{21}$ It works!

Let us use the set 9-17. The difference $d$ is 8. We add 8 to each number in the basic 1-9 magic square: $8+8 \quad 1+8 \quad 6+8$ $3+8 \quad 5+8 \quad 7+8$ $4+8 \quad 9+8 \quad 2+8$

This gives us the new magic square: $16 \quad 9 \quad 14$ $11 \quad 13 \quad 15$ $12 \quad 17 \quad 10$

Let us check its magic sum. The new magic sum should be $15 + (3 \times d)$ . $15 + (3 \times 8)$ $= 15 + 24$ $= \mathbf{39}$ Let us check one row of this square: $16 + 9 + 14$ $= 25 + 14$ $= \mathbf{39}$ It works perfectly!

Answer

To create a magic square using any set of 9 consecutive numbers:

(i) Identify the middle number of your set of 9 consecutive numbers. (ii) Find the difference between your middle number and 5 (the middle number of the 1-9 magic square). Let us call this difference $d$ . (iii) Add this difference $d$ to every number in the classic 1-9 magic square to form your new magic square. The magic sum for your new square will be $15 + (3 \times d)$ .

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