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#1 2007-11-24 05:00:29
- tony123
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- Registered: 2007-08-03
- Posts: 229
one thousand rooms
There are one thousand rooms in the sultan's palace. In each room there is a switch that switches all the lamps in the room on or off. When the lamps were on in each room and the sultan was bored, he walked through all his rooms one by one and repeated his walk again and again, always starting with the first room. During the first walk, he turned all the switches. The second time he turned the switch in every second room. The third time he turned the switch in every third room, and so on. (He turned the light on if it was off, and he turned it off if it was on). When he had walked through his room 500 times, he got tired of the game and decided to go to bed. He needed a room in which the lights were off. Which rooms did he have to choose from?
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#2 2007-11-24 05:08:46
- JaneFairfax
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- Registered: 2007-02-23
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Re: one thousand rooms
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#3 2007-11-24 05:26:05
- mathsyperson
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- Registered: 2005-06-22
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Re: one thousand rooms
This puzzle isn't exactly the same as the linked one though. There's a different amount of rooms, the procedure for switch-switching is slightly different (because he always switches the first one in this case) and he stops the game early.
They both use the same kind of reasoning though, you just need to tweak the thought process a bit.
Why did the vector cross the road?
It wanted to be normal.
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#4 2007-11-24 05:38:56
- JaneFairfax
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- Registered: 2007-02-23
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Re: one thousand rooms
What the hell is so different? Of course the two puzzles are not identical, but they're not worlds apart either! 
The answer is that all those rooms whose numbers have an odd number of positive-integer factors ≤ 500 will have their lights off, and all the others will have their lights on. Thus, those rooms with lights off will be the ones whose numbers are all the perfect squares < 500, and all the integers between 501 and 1000 (inclusive) that are not perfect squares.
Last edited by JaneFairfax (2007-11-24 06:16:45)
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