Question
Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?
✓
Answer
He noticed that the number of toggles a locker receives equals the number of factors of that locker’s number. A locker toggled an odd number of times will be open at the end; a locker toggled an even number of times will be closed. Only perfect squares have an odd number of factors, so exactly the lockers numbered with perfect squares (1, 4, 9, 16, …) remain open.
| Locker Number | Factors | Number of factors/Number of locker toggles |
| 1 | 1(o) | 1 (odd) |
| 2 | 1(o), 2(c) | 2 (even) |
| 3 | 1(o), 3(c) | 2 (even) |
| 4 | 1(o), 2(c), 4(o) | 3 (odd) |
| 5 | 1(o), 5(c) | 2 (even) |
| 6 | 1(o), 2(c), 3(o), 6(c) | 4 (even) |
| 7 | 1(o), 7(c) | 2 (even) |
| 8 | 1(o), 2(c), 4(o), 8(c) | 4 (even) |
| 9 | 1(o), 3(c), 9(o) | 3 (odd) |
| 10 | 1(o), 2(c), 5(o), 10(c) | 4 (even) |
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