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        riddles >> easy >> Electrical switches
        
(Message started by: Altamira_64 on Jan 7^th, 2016, 1:34am)

Title: Electrical switches
Post by Altamira_64 on Jan 7^th, 2016, 1:34am

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?

Title: Re: Electrical switches
Post by rmsgrey on Jan 7^th, 2016, 10:11am

5 at a time: [hide]3[/hide] moves
[hideb]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[/hideb]

6 at a time: [hide]!![/hide] moves
[hideb]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)[/hideb]

Title: Re: Electrical switches
Post by Altamira_64 on Jan 7^th, 2016, 12:24pm

Excellent!
How do you deduce the 9+2n condition?

Title: Re: Electrical switches
Post by towr on Jan 7^th, 2016, 12:37pm

[hide]You need to turn on a switch 9 times, and any more switching needs to cancel out so must be even.[/hide]

Title: Re: Electrical switches
Post by rmsgrey on Jan 8^th, 2016, 1:18pm

on 01/07/16 at 12:24:54, 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.