To solve the problem of determining which doors remain open after all 1000 persons have changed the status of the doors, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Problem**:
- We have 1000 doors, all initially closed.
- Each person toggles the state of certain doors based on their number. For example, `P1` toggles all doors, `P2` toggles every 2nd door, `P3` toggles every 3rd door, and so on up to `P1000`.
2. **Identifying the Toggling Pattern**:
- A door `D_n` will be toggled by every person whose number is a divisor of `n`. For example, door `D6` will be toggled by `P1`, `P2`, `P3`, `P6`.
3. **Counting the Toggles**:
- A door will end up being open if it is toggled an odd number of times. This occurs when the number of divisors of the door number is odd.
4. **Understanding Divisors**:
- A number has an odd number of divisors if and only if it is a perfect square. This is because divisors generally come in pairs (e.g., for `12`, the pairs are `(1, 12)`, `(2, 6)`, `(3, 4)`), but a perfect square has one unpaired divisor (e.g., for `9`, the pairs are `(1, 9)`, `(3, 3)`).
5. **Finding Perfect Squares**:
- We need to find all perfect squares between `1` and `1000`. The perfect squares are `1^2, 2^2, 3^2, ..., n^2` where `n^2 ≤ 1000`.
- The largest integer `n` such that `n^2 ≤ 1000` is `31` because `31^2 = 961` and `32^2 = 1024` which exceeds `1000`.
6. **Listing the Open Doors**:
- The perfect squares from `1` to `31` are:
- `1^2 = 1`
- `2^2 = 4`
- `3^2 = 9`
- `4^2 = 16`
- `5^2 = 25`
- `6^2 = 36`
- `7^2 = 49`
- `8^2 = 64`
- `9^2 = 81`
- `10^2 = 100`
- `11^2 = 121`
- `12^2 = 144`
- `13^2 = 169`
- `14^2 = 196`
- `15^2 = 225`
- `16^2 = 256`
- `17^2 = 289`
- `18^2 = 324`
- `19^2 = 361`
- `20^2 = 400`
- `21^2 = 441`
- `22^2 = 484`
- `23^2 = 529`
- `24^2 = 576`
- `25^2 = 625`
- `26^2 = 676`
- `27^2 = 729`
- `28^2 = 784`
- `29^2 = 841`
- `30^2 = 900`
- `31^2 = 961`
### Conclusion:
The doors that remain open are the doors numbered:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.
