The Locker Problem

The solution to this problem is 31 lockers remain open. The lockers that stayed open are 1,4,9,16,25,36,49,64,81 and so on. We know because these are perfect squares. There is a pattern that goes from an odd number to an even one. Each locker gets changed twice for each set of factors. The perfect squares are the only numbers that have an odd number of factors. If a locker is touched an odd number of times, it will be open. This depends on how many students touched it . A locker touched an even number of times, it will be left closed. Therefore, only the lockers with an odd number of factors will be left open leaving the perfect squares. Perfect squares have a number of factors because factors come in pairs and the square root of a perfect square is only one factor. This graph shown below gives an idea of the pattern of the problem and how we got the solution. It gives you and idea of how the pattern works.