Take a magic square, and

(a) increase each number by 1

(b) double each number

In each case, is the resulting grid also a magic square? How do the magic sums change in each case?

Question

Take a magic square, and (a) increase each number by 1 (b) double each number In each case, is the resulting grid also a magic square? How do the magic sums change in each case?

Homi · Accepted Answer

We will check if a magic square remains magic after changing its numbers.

Step 1 — Understand the Original Magic Square

First, let us write down the given magic square. A magic square has the same sum for all rows, columns, and main diagonals. Let us find this magic sum for the given square.

The original magic square is: 8 1 6 3 5 7 4 9 2

We will add the numbers in each row. Sum of Row 1: $8 + 1 + 6$ $= 15$ Sum of Row 2: $3 + 5 + 7$ $= 15$ Sum of Row 3: $4 + 9 + 2$ $= 15$

Next, we will add the numbers in each column. Sum of Column 1: $8 + 3 + 4$ $= 15$ Sum of Column 2: $1 + 5 + 9$ $= 15$ Sum of Column 3: $6 + 7 + 2$ $= 15$

Now, we will add the numbers in the main diagonals. Sum of Diagonal 1 (top-left to bottom-right): $8 + 5 + 2$ $= 15$ Sum of Diagonal 2 (top-right to bottom-left): $6 + 5 + 4$ $= 15$

All sums are 15. So, the original magic sum is 15.

$\boxed{\text{Original Magic Sum} = 15}$

Step 2 — Increase Each Number by 1

We will add 1 to every number in the original square. Let us create the new grid.

The new grid is: 8+1 1+1 6+1 -> 9 2 7 3+1 5+1 7+1 -> 4 6 8 4+1 9+1 2+1 -> 5 10 3

The new square is: 9 2 7 4 6 8 5 10 3

Now, we will check if this new grid is a magic square. We will find the sum of each row. Sum of Row 1: $9 + 2 + 7$ $= 18$ Sum of Row 2: $4 + 6 + 8$ $= 18$ Sum of Row 3: $5 + 10 + 3$ $= 18$

Next, we will find the sum of each column. Sum of Column 1: $9 + 4 + 5$ $= 18$ Sum of Column 2: $2 + 6 + 10$ $= 18$ Sum of Column 3: $7 + 8 + 3$ $= 18$

Finally, we will find the sum of the main diagonals. Sum of Diagonal 1: $9 + 6 + 3$ $= 18$ Sum of Diagonal 2: $7 + 6 + 5$ $= 18$

All sums are 18. So, the new grid is also a magic square. The new magic sum is 18. The original sum was 15. The new sum is $18 - 15 = 3$ more. This is because we added 1 to each of the three numbers in every row, column, and diagonal. So, the sum increased by $3 \times 1 = 3$ .

$\boxed{\text{New Magic Sum} = 18}$

Step 3 — Double Each Number

We will multiply every number in the original square by 2. Let us create this new grid.

The new grid is: 82 12 62 -> 16 2 12 32 52 72 -> 6 10 14 42 92 2*2 -> 8 18 4

The new square is: 16 2 12 6 10 14 8 18 4

Now, we will check if this new grid is a magic square. We will find the sum of each row. Sum of Row 1: $16 + 2 + 12$ $= 30$ Sum of Row 2: $6 + 10 + 14$ $= 30$ Sum of Row 3: $8 + 18 + 4$ $= 30$

Next, we will find the sum of each column. Sum of Column 1: $16 + 6 + 8$ $= 30$ Sum of Column 2: $2 + 10 + 18$ $= 30$ Sum of Column 3: $12 + 14 + 4$ $= 30$

Finally, we will find the sum of the main diagonals. Sum of Diagonal 1: $16 + 10 + 4$ $= 30$ Sum of Diagonal 2: $12 + 10 + 8$ $= 30$

All sums are 30. So, the new grid is also a magic square. The new magic sum is 30. The original sum was 15. The new sum is $30 \div 15 = 2$ times the original sum. This is because we multiplied each number by 2. So, the sum of three numbers also gets multiplied by 2.

$\boxed{\text{New Magic Sum} = 30}$

Answer

(a) Yes, the resulting grid is still a magic square. The new magic sum is 18, which increases by 3 from the original sum. (b) Yes, the resulting grid is still a magic square. The new magic sum is 30, which is double the original sum.

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