The locker problem

 Lockers are only touched by students who are factors of that locker number. For example, locker #5 is only touched by students 1 and 5. Student 1 closes it and student 5 opens it. In fact, because factors come in pairs, the first factor student closes it and the corresponding factor student opens it. When factors don’t come in pairs, the locker will be left close. And factors don’t come in pairs when numbers are multiplied by themselves. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100,…) are the only numbers whose factors don’t come in pairs because one set of factors, the square root, is multiplied by itself. This means that only perfect square lockers will be left close.

Therefore, locker # 1, 4, 9, 16, 25, 36,…, 961 these perfect squares numbers are left close whereas the rest of lockers are left open!