Question

An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of

every column, and of the two diagonals is the same. For which n is there a magic square whose
entries are consecutive positive integers with a sum of 2015? For each such n, what are the
largest and smallest entries in the square? ​​​​​​​

Answer 1

Consider the given conditions regarding magic square and find the smallest and largest entries in the square along with the value of n.It is given that the sum of every row, of every column, and of the two diagonals of a magic square is the same. Let 's' be the sum of each row, column, and diagonal. Then, the sum of n numbers in the row, column, or diagonal is s.Therefore, the sum of all the numbers in the magic square will be n × s. Given, n > 1, and the sum of the entries is 2015. Therefore, s is a factor of 2015, which has the prime factorization $5\cdot13\cdot31$.The magic sum s must be the product of three distinct factors of 2015. As there are only three distinct prime factors of 2015, s must be $5\cdot13\cdot31=2015$. Thus, the sum of each row, column, and diagonal is 2015.Now, to determine the value of n, we can solve the equation, $2015=n\cdot\dfrac{n^2+1}{2}$.On solving this equation, we get n = 5. Thus, we can form a 5 x 5 magic square with the entries being the consecutive positive integers. In this case, the smallest number will be 243 and the largest number will be 287.