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There are 1000 doors D1;D2; : : : ;D1000 and 1000 persons P1; P2; : : : ; P1000. Initially all the doors were closed. Person P1 goes and opens all the doors. Then person P2 closes door D2;D4; : : : ;D1000 and leaves the odd num- bered doors open. Next P3 changes the state of every third door, that is, D3;D6; : : :D999. (For instance, P3 closes the open door D3 and opens the closed door D6, and so on). Similarly, Pm changes the state of the the doors Dm;D2m;D3m; : : : ;Dnm; : : : while leaving the other doors un- touched. Finally, P1000 opens D1000 if it was closed or closes it if it were open. At the end, how many doors will remain open?

There are 1000 doors D1;D2; : : : ;D1000 and 1000 persons P1; P2; : : : ; P1000.
Initially all the doors were closed. Person P1 goes and opens all the doors.
Then person P2 closes door D2;D4; : : : ;D1000 and leaves the odd num-
bered doors open. Next P3 changes the state of every third door, that
is, D3;D6; : : :D999. (For instance, P3 closes the open door D3 and opens
the closed door D6, and so on). Similarly, Pm changes the state of the
the doors Dm;D2m;D3m; : : : ;Dnm; : : : while leaving the other doors un-
touched. Finally, P1000 opens D1000 if it was closed or closes it if it were
open. At the end, how many doors will remain open?

Grade:12

1 Answers

Raj
19 Points
8 years ago

Sir my answer is  32 doors are  in open

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