In the 3 x 3 square on the left, the numbers from 1 to 9 have been filled so that the sum in each row/column/diagonal adds up to 15. If the same exercise is to be carried out for the 4 x 4 square, using the numbers from 1 to 16, what will be the sum of the numbers in each row / column / diagonal?
Show Hint
Memorizing the formula for the magic constant, \( M = \frac{n(n^2 + 1)}{2} \), is a very efficient way to solve problems involving magic squares in competitive exams.
Step 1: Understanding the Question:
The question describes a magic square. A magic square of order 'n' is an n x n grid filled with distinct integers from 1 to n², such that the sum of the integers in each row, column, and main diagonal is the same. This sum is called the "magic constant". We need to find the magic constant for a 4 x 4 magic square. Step 2: Key Formula or Approach:
The formula to calculate the magic constant (M) for a normal magic square of order n is:
\[ M = \frac{n(n^2 + 1)}{2} \]
Alternatively, you can find the sum of all the numbers in the grid and divide it by the order of the square (n). The sum of the first 'k' integers is given by \( \frac{k(k+1)}{2} \). Step 3: Detailed Explanation: Method 1: Using the Magic Constant Formula
For a 4 x 4 square, the order is n = 4.
Plugging n = 4 into the formula:
\[ M = \frac{4(4^2 + 1)}{2} = \frac{4(16 + 1)}{2} = \frac{4(17)}{2} = 2 \times 17 = 34 \]
Method 2: Using the Sum of Numbers
The numbers used are from 1 to 16. The total sum of these numbers is:
\[ \text{Sum} = \frac{16(16 + 1)}{2} = \frac{16 \times 17}{2} = 8 \times 17 = 136 \]
Since there are 4 rows (or columns) and the sum of each must be the same, we divide the total sum by 4.
\[ \text{Sum per row} = \frac{\text{Total Sum}}{4} = \frac{136}{4} = 34 \]
Both methods yield the same result. Step 4: Final Answer:
The sum of the numbers in each row, column, and diagonal for the 4 x 4 square will be 34.
