The Locker Problem Solution

The Locker Problem

There are 1000 lockers, and if the first student opens all of the lockers, the second closes every other, the third changes the state of every third, and so on, there are going to be 31 lockers left open and969 closed. The lockers left open will be 1, 4, 9, 16, 25, 36, 49, 64, etc. until we reach 961. All of the numbers that we listed are perfect squares meaning that a number is multiplied by itself to get that number, for example 1², 2², 3², 4², etc. The reason why only the perfect squares are left open is because they have an odd number of factors. So the locker is going to get open, close, opened, or open, close, open, close, opened, or however long it takes. I’ll use 16 and 18 as examples and will illustrate them in the diagram below. 18 has the factors, 1, 2, 3, 6, 9, and 18. Every one of these factors has a pair, 1x18, 2x9, 3x6, so it goes, open, close, open, close, open, closed. 16 has the factors 1,2,4,8, and 16. The pairs are 1x16, 2x8, but 4 is left by itself, it is the square root of 16. This is why the square numbers are left open; they have that square root, giving them an odd number of factors. There was another pattern that was found, but the flaw was that it would have taken too long to measure on a large scale. The pattern, staring with locker one, was 1 open, 2 closed, 1 open, 4 closed, 1 open, 6 closed, 1 open, 8 closed, etc.

If the diagram seems confusing, here is how to read it. The first row shows the locker #, below that shows the first student opening every locker, below that shows the next person who touched that locker and if they open or closed it. If there is a c next to/below the number, that student closed it, same for o except open instead of close. If there is nothing in the box, then it won’t be touched anymore and you know whether it is open or closed

Locker #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

1o

2c

3c

2c

5c

2c

7c

2c

3c

2c

11c

2c

13c

2c

3c

2c

17c

2c

19c

2c

4o

3o

4o

9o

5o

3o

7o

5o

4o

3o

4o

6c

8c

10c

4c

14c

15c

8c

6c

5c

6o

16o

9o

10o

12c

18c

20c