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Which doors are open in the end?

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There are 100 doors in a row, all doors are initially closed. A person walks through all doors multiple times and toggle (if open then close, if close then open) them in following way:

In first walk, the person toggles every door.

In second walk, the person toggles every second door, i.e., 2nd, 4th, 6th, 8th, …

In third walk, the person toggles every third door, i.e. 3rd, 6th, 9th, …

………
……….

In 100th walk, the person toggles 100th door.

Which doors are open in the end?

posted Jul 24, 2017 by Sumanta Hazra

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2 Answers

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Ten doors are open in the end.
1st, 4th, 9th, 16th, 25th, 36th, 49th, 64th, 81th and 100th doors(perfect square number doors) are open.

answer Jul 25, 2017 by 이기가
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Solution:
A door is toggled in i th walk if i divides door number. For example the door number 45 is toggled in 1st, 3rd, 5th, 9th and 15th walk.
The door is switched back to initial stage for every pair of divisors. For example 45 is toggled 6 times for 3 pairs (5, 9), (15, 3) and (1, 45).
It looks like all doors would become closes at the end. But there are door numbers which would become open, for example 16, the pair (4, 4) means only one walk. Similarly all other perfect squares like 4, 9, ….

So the answer is 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

answer Sep 21, 2017 by Mogadala Ramana



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enter image description here

There are 13 caves arranged in a circle. There is a thief hiding in one of the caves. Each day the the thief can move to any one of of the caves that is adjacent to the cave in which he was staying the previous day. And each day, you are allowed to enter any two caves of your choice.

What is the minimum number of days to guarantee in which you can catch the thief?

Note: Thief may or may not move to adjacent cave. You can check any two caves, not necessarily be adjacent. If thief and you exchange your caves, you will surely cross at some point, and you can catch the thief immediately.

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