Magic Square


Magic Square


The MATHCOUNTS Foundation is a 501(c)(3) non-profit organization that strives to engage students of all ability and interest levels in fun, challenging math programs, in order to expand their academic and professional opportunities. Middle school students exist at a critical juncture in which their love for mathematics must be nurtured, or their fear of mathematics must be overcome. MATHCOUNTS provides students with the kinds of experiences that foster growth and transcend fear to lay a foundation for future success.

For more than 30 years MATHCOUNTS has provided enriching, extracurricular opportunities to students and free, high-quality resources to educators. Every child is unique, but we believe all children are capable of seeing the beauty and joy of math, whether they come to us already passionate about math, or intimidated by it.

There are many paths to math. We work to ensure that all students discover theirs.

Magic Square

The magic square shown contains each of the positive integers 1-25. The sum of the numbers in each row, each column and both diagonals is the same. What is the value of n?

Let’s start by finding the sum of the positive integers 1, 2, 3, …, 25. Notice we can pair up the integers into sums of 26: 1 and 25, 2 and 24, 3 and 23, and so on. There are 12 of these pairs, with the integer 13 left in the middle with no partner. The sum of the 25 integers, then, is 26 × 12 + 13 = 312 + 13 = 325. Because each of the five rows must have the same sum, we can determine that sum to be 325 ÷ 5 = 65. Adding the known values in the 4th row, we get 10 + 12 + 21 + 3 = 46. Therefore, the value of n is 65 − 46 = 19