Create a magic square using the numbers 2 – 10. What strategy would you use for this? Compare it with the magic squares made using 1 – 9.

Question

Homi · Accepted Answer

A magic square is a grid where numbers in each row, column, and main diagonal add up to the same total.

Step 1 — Finding the 1-9 Magic Square

We first need a magic square using numbers from 1 to 9. This is a standard 3x3 magic square. The numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9. The middle number of this set is 5. This middle number usually goes in the center of the square. Let us write down the classic 1-9 magic square.

Diagram 1

The magic square for numbers 1 to 9 is: 8, 1, 6 3, 5, 7 4, 9, 2

Now, let us find the magic sum for this square. We can add numbers in any row, column, or diagonal. Let us add the numbers in the first row.

$8 + 1 + 6$

$= 9 + 6$

$\boxed{15}$

So, the magic sum for the 1-9 square is 15.

Step 2 — Strategy for 2-10 Magic Square

We need to create a magic square using numbers from 2 to 10. These numbers are 2, 3, 4, 5, 6, 7, 8, 9, 10. Notice that each of these numbers is exactly 1 more than the numbers 1 to 9. For example, 2 is $1+1$ , 3 is $2+1$ , and so on. The simplest strategy is to use our 1-9 magic square. We will add 1 to every number in that magic square. This will give us a new magic square. It will use the numbers 2 to 10.

Step 3 — Creating the 2-10 Magic Square

Let us take each number from the 1-9 magic square. We will add 1 to each number.

Original 1-9 square: 8, 1, 6 3, 5, 7 4, 9, 2

Applying the strategy (adding 1 to each number): $8+1=9$ , $1+1=2$ , $6+1=7$ $3+1=4$ , $5+1=6$ , $7+1=8$ $4+1=5$ , $9+1=10$ , $2+1=3$

The new magic square for numbers 2 to 10 is: 9, 2, 7 4, 6, 8 5, 10, 3

Let us find the magic sum for this new square. We can add numbers in any row, column, or diagonal. Let us add the numbers in the first row.

$9 + 2 + 7$

$= 11 + 7$

$\boxed{18}$

So, the magic sum for the 2-10 square is 18.

Diagram 2

Step 4 — Comparison of Magic Squares

We have two magic squares now. One uses numbers 1 to 9. The other uses numbers 2 to 10.

Let us compare their features.

The magic sum for the 1-9 square is 15. The magic sum for the 2-10 square is 18. The new magic sum is $15 + 3 = 18$ . This is because we added 1 to each of the three numbers in every row, column, and diagonal.

The structure of both squares is the same. The relative positions of the numbers are identical. For example, the center number in both squares is the middle number of their set. For 1-9, the center is 5. For 2-10, the center is 6. And 6 is $5+1$ . Every number in the 2-10 square is exactly 1 more than the corresponding number in the 1-9 square.

Answer

(i) Strategy: Start with the classic magic square using numbers 1 to 9, then add 1 to each number in that square. (ii) New Magic Square (2-10): 9, 2, 7 4, 6, 8 5, 10, 3 (iii) Comparison: The magic square for numbers 2-10 has a magic sum of 18. The magic square for numbers 1-9 has a magic sum of 15. The structure and relative positions of numbers are the same. Each number in the 2-10 square is 1 greater than the corresponding number in the 1-9 square.

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