Question 6

A magic square uses numbers 1-9 where every row, column, and main diagonal adds up to the same number.

Step 1 — Find the Magic Constant

Let us find the sum of all numbers from 1 to 9. We add them all up.

$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9$

$= 45$

Let us call this total sum S. So, $S = 45$. A 3x3 magic square has 3 rows. Each row must add up to the same number. So, the sum of all numbers (S) must be shared equally among the 3 rows. This special sum is called the magic constant. Let us call the magic constant M.

$M = S \div 3$

$M = 45 \div 3$

$\boxed{M = 15}$

So, every row, column, and main diagonal in our magic square must add up to 15.

Step 2 — Find the Center Number

Let us think about the number in the very middle of the square. Let us call this number 'e'. This middle number is special. It is part of the middle row, the middle column, and both diagonal lines. So, it is used in four different sums. All other numbers are used in fewer sums. We can find 'e' using a clever trick. The sum of the middle row, middle column, and both diagonals is $4 \times M$. This sum also equals the total sum of all numbers (S) plus three times the center number (3e).

$4 \times M = S + 3 \times e$

We know $M = 15$ and $S = 45$.

$4 \times 15 = 45 + 3 \times e$

$60 = 45 + 3 \times e$

Now, we subtract 45 from both sides.

$60 - 45 = 3 \times e$

$15 = 3 \times e$

To find 'e', we divide 15 by 3.

$e = 15 \div 3$

$\boxed{e = 5}$

The number in the very center of the magic square must be 5.

Step 3 — Find the Corner Numbers

We have used the number 5. The remaining numbers are 1, 2, 3, 4, 6, 7, 8, 9. Let us think about the four corner numbers. Consider a corner number, for example, the top-left one. The number opposite to it, through the center (5), must add up to 10 with it. For example, if the top-left is 1, the bottom-right must be 9 (because $1+5+9=15$). The pairs that add up to 10 are (1,9), (2,8), (3,7), (4,6).

Let us try to put odd numbers in the corners. The odd numbers (excluding 5) are 1, 3, 7, 9. Let us place 1 in the top-left corner and 7 in the top-right corner. The top row must add up to 15. So, $1 + \text{middle top number} + 7 = 15$. This means the middle top number must be $15 - 1 - 7 = 7$. But the number 7 is already used as a corner number. Each number can only be used once in a magic square. This means our assumption was wrong. The corner numbers cannot be the odd numbers (1, 3, 7, 9). Therefore, the corner numbers must be the even numbers: 2, 4, 6, 8. The numbers in the middle of each side must be the remaining odd numbers: 1, 3, 7, 9.

Step 4 — Construct the Unique Magic Square

We know the center is 5. The corners are 2, 4, 6, 8. The middle-side numbers are 1, 3, 7, 9. Let us place these numbers.

1. Place 5 in the center.
2. Place 2 in the top-left corner.
3. The number opposite to 2 (through 5) must be $10 - 2 = 8$. Place 8 in the bottom-right corner.
4. Place 4 in the top-right corner.
5. The number opposite to 4 (through 5) must be $10 - 4 = 6$. Place 6 in the bottom-left corner.

Now, let us fill the remaining empty spots. Look at the top row: $2 + \text{empty} + 4 = 15$. The empty spot must be $15 - 2 - 4 = 9$. Place 9 in the top-middle spot.

Look at the left column: $2 + \text{empty} + 6 = 15$. The empty spot must be $15 - 2 - 6 = 7$. Place 7 in the middle-left spot.

Look at the middle row: $7 + 5 + \text{empty} = 15$. The empty spot must be $15 - 7 - 5 = 3$. Place 3 in the middle-right spot.

Look at the middle column: $9 + 5 + \text{empty} = 15$. The empty spot must be $15 - 9 - 5 = 1$. Place 1 in the bottom-middle spot.

This gives us the magic square:

Let us check all sums: Rows: $2+9+4=15$, $7+5+3=15$, $6+1+8=15$. (Correct!) Columns: $2+7+6=15$, $9+5+1=15$, $4+3+8=15$. (Correct!) Diagonals: $2+5+8=15$, $4+5+6=15$. (Correct!)

This is the only way to arrange the numbers 1-9 into a 3x3 magic square, without counting turns or flips as different. So, there is exactly 1 unique magic square.

Step 5 — Count Variations with Rotations and Reflections

A square shape has 8 different ways it can be positioned. Imagine our unique magic square is on a piece of paper. We can turn the paper in 4 ways:

1. The original square.
2. Turn it 90 degrees clockwise.
3. Turn it 180 degrees.
4. Turn it 270 degrees clockwise. These are 4 different magic squares.

We can also flip the paper over. There are 4 ways to flip a square: 5. Flip the original square across a vertical line (left becomes right). 6. Flip the original square across a horizontal line (top becomes bottom). 7. Flip the original square across its main diagonal (top-left to bottom-right). 8. Flip the original square across its anti-diagonal (top-right to bottom-left).

Each of these 8 actions creates a different arrangement of numbers that is still a valid magic square. All these 8 arrangements are distinct from each other. So, if we allow rotations and reflections, there are 8 variations of this magic square.

Answer

(i) There is exactly 1 unique magic square using the numbers 1-9 (excluding rotations and reflections). (ii) If transformations like rotations and reflections are allowed, there are 8 variations of this magic square.