The Locker Problem

The Locker Problem

To know which lockers are opened and which are closed is all based on square roots and factors. Every number that has a perfect square root is open. For example, 49 is an open locker. This is an open locker because it is a square root number. The number 49 comes from 7 x 7 = 49. Just because the number 49 is a perfectly squared number does not mean taht there aren't other square root numbers in the Locker Problem. Within the first 100 lockers there are exactly 10 squared numbers. Those numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Why? They are all like 49 where you find them by using the square roots. In the end, this leaves us with 31 open lockers and 969 closed lockers whcih make a total of 1000 lockers.

What do factors have to do with all of my shenanigans above? If you listed all the common facotrs of the square root numbers you could have an odd amount of factors. As an example, I will use my dear old 49 again. The number 49 has a total of 3 factors. Those factors are 1, 7, and 49. All of these numbers could translate into Open, Close, and finally Open. This is because the first person opened every single locker. This means that all the lockers' factors all have to start with a 1, which stands for Open. This can mean that every single squared number has an odd number of factors, including the number 1, the number that if they timed it by itself would become the squared number, and finally the squared number itself.

Included is a table of the first 20 lockers in the locker problem.