There are 1000 lockers numbered 1 to 1000. Suppose you open all of the lockers, then close every other locker. Then, for every third locker, you close each opened locker and open each closed locker. You follow this same pattern for every fourth locker, every fifth locker, and so on up to every thousandth locker. Which locker doors will be open when the process is complete?
Consider lockers numbered 1 to 25 to begin. All doors are opened. Then every other door is closed.
| Locker # | Status: | |
| 1 | 2 | |
| 1 | open | |
| 2 | open | closed |
| 3 | open | |
| 4 | open | closed |
| 5 | open | |
| 6 | open | closed |
| 7 | open | |
| 8 | open | closed |
| 9 | open | |
| 10 | open | closed |
| 11 | open | |
| 12 | open | closed |
| 13 | open | |
| 14 | open | closed |
| 15 | open | |
| 16 | open | closed |
| 17 | open | |
| 18 | open | closed |
| 19 | open | |
| 20 | open | closed |
| 21 | open | |
| 22 | open | closed |
| 23 | open | |
| 24 | open | closed |
| 25 | open |
That is, every multiple of 1 is opened and every multiple of 2 is then closed.
Next, every third locker changes from open to closed or closed to open. That is, every locker numbered as a multiple of three, changes its status.
| Locker # | Status: | ||
| 1 | 2 | 3 | |
| 1 | open | ||
| 2 | open | closed | |
| 3 | open | closed | |
| 4 | open | closed | |
| 5 | open | ||
| 6 | open | closed | open |
| 7 | open | ||
| 8 | open | closed | |
| 9 | open | closed | |
| 10 | open | closed | |
| 11 | open | ||
| 12 | open | closed | open |
| 13 | open | ||
| 14 | open | closed | |
| 15 | open | closed | |
| 16 | open | closed | |
| 17 | open | ||
| 18 | open | closed | open |
| 19 | open | ||
| 20 | open | closed | |
| 21 | open | closed | |
| 22 | open | closed | |
| 23 | open | ||
| 24 | open | closed | open |
| 25 | open |
Then, every fourth locker changes from open to closed or closed to open. Or, another way to look at the situation is that every locker numbered as a multiple of four changes its status.
| Locker # | Status: | |||
| 1 | 2 | 3 | 4 | |
| 1 | open | |||
| 2 | open | closed | ||
| 3 | open | closed | ||
| 4 | open | closed | open | |
| 5 | open | |||
| 6 | open | closed | open | |
| 7 | open | |||
| 8 | open | closed | open | |
| 9 | open | closed | ||
| 10 | open | closed | ||
| 11 | open | |||
| 12 | open | closed | open | closed |
| 13 | open | |||
| 14 | open | closed | ||
| 15 | open | closed | ||
| 16 | open | closed | open | |
| 17 | open | |||
| 18 | open | closed | open | |
| 19 | open | |||
| 20 | open | closed | open | |
| 21 | open | closed | ||
| 22 | open | closed | ||
| 23 | open | |||
| 24 | open | closed | open | closed |
| 25 | open |
On the fifth pass, lockers numbered as multiples of five are changed from open to closed or closed to open.
| Locker # | Status: | ||||
| 1 | 2 | 3 | 4 | 5 | |
| 1 | open | ||||
| 2 | open | closed | |||
| 3 | open | closed | |||
| 4 | open | closed | open | ||
| 5 | open | closed | |||
| 6 | open | closed | open | ||
| 7 | open | ||||
| 8 | open | closed | open | ||
| 9 | open | closed | |||
| 10 | open | closed | open | ||
| 11 | open | ||||
| 12 | open | closed | open | closed | |
| 13 | open | ||||
| 14 | open | closed | |||
| 15 | open | closed | open | ||
| 16 | open | closed | open | ||
| 17 | open | ||||
| 18 | open | closed | open | ||
| 19 | open | ||||
| 20 | open | closed | open | closed | |
| 21 | open | closed | |||
| 22 | open | closed | |||
| 23 | open | ||||
| 24 | open | closed | open | closed | |
| 25 | open | closed |
Notice that the locker changes status when every nth locker is open or closed. That is, the locker door will change from open or closed when "n" is a factor of the locker number.
Also notice that since the doors were opened first, that the doors opened and closed an odd number of times remain open when the number of factors of the locker number is odd. That is, the lockers numbered with perfect squares are left open at the end of the process. Perfect squares have an odd number of distinct factors and remain open.
| Locker # | Status: | ||||||||||||||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | |
| 1 | open | ||||||||||||||||||||||||
| 2 | open | closed | |||||||||||||||||||||||
| 3 | open | closed | |||||||||||||||||||||||
| 4 | open | closed | open | ||||||||||||||||||||||
| 5 | open | closed | |||||||||||||||||||||||
| 6 | open | closed | open | closed | |||||||||||||||||||||
| 7 | open | closed | |||||||||||||||||||||||
| 8 | open | closed | open | closed | |||||||||||||||||||||
| 9 | open | closed | open | ||||||||||||||||||||||
| 10 | open | closed | open | closed | |||||||||||||||||||||
| 11 | open | closed | |||||||||||||||||||||||
| 12 | open | closed | open | closed | open | closed | |||||||||||||||||||
| 13 | open | closed | |||||||||||||||||||||||
| 14 | open | closed | open | closed | |||||||||||||||||||||
| 15 | open | closed | open | closed | |||||||||||||||||||||
| 16 | open | closed | open | closed | open | ||||||||||||||||||||
| 17 | open | closed | |||||||||||||||||||||||
| 18 | open | closed | open | closed | open | closed | |||||||||||||||||||
| 19 | open | closed | |||||||||||||||||||||||
| 20 | open | closed | open | closed | open | closed | |||||||||||||||||||
| 21 | open | closed | open | closed | |||||||||||||||||||||
| 22 | open | closed | open | closed | |||||||||||||||||||||
| 23 | open | closed | |||||||||||||||||||||||
| 24 | open | closed | open | closed | open | closed | open | closed | |||||||||||||||||
| 25 | open | closed | open |
Look at the factors of a few numbers and the results of each pass.
36: 1 open; 2 closed; 3 open; 4 closed; 6 open; 9 closed; 12 open; 18 closed; 36 open
49: 1 open; 7 closed; 49 open
55: 1 open; 5 closed; 11 open; 55 closed
64: 1 open; 2 closed; 4 open; 8 closed; 16 open; 32 closed; 64 open
81: 1 open; 3 closed; 9 open; 27 closed; 81 open
100: 1 open; 2 closed; 4 open; 5 closed; 10 open; 20 closed; 25 open; 50 closed; 100 open
125: 1 open; 5 closed; 25 open; 125 closed
275: 1 open; 5 closed; 11 open; 25 closed; 55 open; 275 closed
324: 1 open; 2 closed; 3 open; 4 closed; 6 open; 9 closed; 12 open; 18 closed; 27 open; 36 closed; 54 open; 81 closed; 108 open; 162 closed; 324 open
The doors of lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 will remain open on the first 100 passes through the lockers. Lockers numbered 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961 will also remain open since these numbers are the squares of numbers 11 to 31.
Return to Chris Reid's home page