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Topic: Electrical switches (Read 444 times) |
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Altamira_64
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Electrical switches
« on: Jan 7th, 2016, 1:34am » |
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We have a set of 9 ON-OFF switches on an electrical board, initially all of them at the OFF position. What is the minimum number of moves to put them all at the ON position, if we must switch exactly 5 of them at each move? Same question but for 6 switches in each move?
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rmsgrey
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Re: Electrical switches
« Reply #1 on: Jan 7th, 2016, 10:11am » |
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5 at a time: 3 moves hidden: | Parity considerations say it must be an odd number of moves. You need a total of 9+2n switchings, for n non-negative, so the minimum is 3 moves. It's easy to find solutions that work in 3 moves - eg 12345, 12367, 12389 | 6 at a time: !! moves hidden: | You need a total of 9+2n switchings, which is odd, but each move makes an even number of switchings. Since there are no numbers which are both odd and even, parity considerations mean it can't be done (at least not with a finite number of moves) |
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Altamira_64
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Re: Electrical switches
« Reply #2 on: Jan 7th, 2016, 12:24pm » |
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Excellent! How do you deduce the 9+2n condition?
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towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
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Re: Electrical switches
« Reply #3 on: Jan 7th, 2016, 12:37pm » |
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You need to turn on a switch 9 times, and any more switching needs to cancel out so must be even.
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« Last Edit: Jan 7th, 2016, 12:38pm by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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rmsgrey
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Re: Electrical switches
« Reply #4 on: Jan 8th, 2016, 1:18pm » |
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on Jan 7th, 2016, 12:24pm, Altamira_64 wrote:Excellent! How do you deduce the 9+2n condition? |
| You need 9 more "on"s than "off"s. n offs plus (n+9) ons makes 2n+9 total switchings.
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