There are 31 lockers open and 969 are shut. This is because the lockers that are open are perfect squares. They are perfect squares because they have an odd amount of factors, The amount of factors a number has corresponds to the amount of times a locker’s state is changed. As an example, 8’s factors are 1, 2, 4, and 8. That means that the first, second, fourth, and eighth student will touch locker 8. A perfect square is the only type of number with an odd amount of factors because you only list its square root as one number. As an example, 16's factors are, 1, 2, 4, 8, and 16. 4 is the square root of 16 so it is only listed once.
The reason that the lockers do not change once their number is passed is because after the 16th kid finishes the 16th locker, no other kid from then on will touch the 16th locker. that is why when you look at the diagram below that the first locker stays the same and the second locker stays the same after the second kid. This is the same all no matter how high your numbers reach.
There is also a pattern to this: One locker is open, then the next two are shut, one open, four shut, one open, six shut, one open, eight shut...
Also there is an Interactive diagram at the bottom of the page. You should check it out.
Lockers 1-9 | ||||||||||
| K I D S 1 - 9 | O | O | O | O | O | O | O | O | O | |
| O | C | O | C | O | C | O | C | O | ||
| O | C | C | C | O | O | O | C | C | ||
| O | C | C | O | O | O | O | O | C | ||
| O | C | C | O | C | O | O | O | C | ||
| O | C | C | O | C | C | O | O | C | ||
| O | C | C | O | C | C | C | O | C | ||
| O | C | C | O | C | C | C | C | C | ||
| O | C | C | O | C | C | C | C | O | ||
| O=OPEN | C=CLOSED | |||||||||
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