Using this generalised form, find a magic square if the centre number is 25.

Question

Homi · Accepted Answer

We use a special pattern with a given center number to build a magic square.

Step 1 — Understand the Pattern

We have a general way to write numbers in a 3x3 grid. This pattern uses a letter 'm' for the center number. The pattern tells us how to find all other numbers. We will write down the pattern for each cell.

Row 1 cells are: $m + 3$ , $m - 4$ , $m + 1$ . Row 2 cells are: $m - 2$ , $m$ , $m + 2$ . Row 3 cells are: $m - 1$ , $m + 4$ , $m - 3$ .

Step 2 — Use the Center Number

The problem tells us the center number is 25. So, we know that the letter 'm' is equal to 25. We will put 25 in place of 'm' in our pattern.

Step 3 — Calculate Each Number

Now we will do the sums and differences for each cell. Let us find the numbers for the first row.

First cell of Row 1: $m + 3$ $= 25 + 3$

$\boxed{28}$

Second cell of Row 1: $m - 4$ $= 25 - 4$

$\boxed{21}$

Third cell of Row 1: $m + 1$ $= 25 + 1$

$\boxed{26}$

Let us find the numbers for the second row.

First cell of Row 2: $m - 2$ $= 25 - 2$

$\boxed{23}$

Second cell of Row 2: $m$ $= 25$

$\boxed{25}$

Third cell of Row 2: $m + 2$ $= 25 + 2$

$\boxed{27}$

Let us find the numbers for the third row.

First cell of Row 3: $m - 1$ $= 25 - 1$

$\boxed{24}$

Second cell of Row 3: $m + 4$ $= 25 + 4$

$\boxed{29}$

Third cell of Row 3: $m - 3$ $= 25 - 3$

$\boxed{22}$

Step 4 — Build the Magic Square

We now have all nine numbers for our square. We will arrange them into the 3x3 grid. The first row numbers are 28, 21, 26. The second row numbers are 23, 25, 27. The third row numbers are 24, 29, 22.

Diagram 1

Step 5 — Find the Magic Sum

In a magic square, all rows, columns, and diagonals add up to the same number. This number is called the magic sum. Let us check the sum of the first row. $28 + 21 + 26$ $= 49 + 26$

$\boxed{75}$

This is our magic sum. All other rows, columns, and diagonals will also add up to 75.

Answer

(i) The first row of the magic square is: 28, 21, 26 (ii) The second row of the magic square is: 23, 25, 27 (iii) The third row of the magic square is: 24, 29, 22 (iv) The magic sum is: 75

Question 11

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