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Topic: Longest Binary Number (Read 3711 times) |
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ThudnBlunder
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Longest Binary Number
« on: Apr 21st, 2007, 6:58pm » |
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What is the longest binary number which does not contain any 3-bit sequence more than twice or the sequence 11?
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Re: Longest Binary Number
« Reply #1 on: Apr 22nd, 2007, 1:04pm » |
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? 1010101010101010101....?
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towr
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Re: Longest Binary Number
« Reply #2 on: Apr 22nd, 2007, 1:27pm » |
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on Apr 22nd, 2007, 1:04pm, tiber13 wrote:| ? 1010101010101010101....? |
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That contains the sequence 101 and 010 a lot more than twice.
There are 8 three bit sequences, two contain 11, so that leaves 6. Each can occur at most twice, and they overlap with two. So at most the string is 14 characters long.
Another more fruitfull approach is to consider we must always have one or two 0's between every two 1's. But then any internal 1 will be part of the sequence 010, and we can have only two of those.
Tentatively, 101001001
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| « Last Edit: Apr 22nd, 2007, 1:31pm by towr » |
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Re: Longest Binary Number
« Reply #3 on: Apr 22nd, 2007, 1:31pm » |
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? sorry, but i dont understand binary, but i do know statistics, 5/4 people are terrible at fractions
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towr
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Re: Longest Binary Number
« Reply #4 on: Apr 22nd, 2007, 1:33pm » |
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on Apr 22nd, 2007, 1:31pm, tiber13 wrote:| ? sorry, but i dont understand binary, but i do know statistics, 5/4 people are terrible at fractions |
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Hmm, yes, I always forget there are 10 kinds of people: those that understand binary and those that don't. 
Instead of binary, you can consider decimal (i,.e. regular) numbers which only have 1's and 0's.
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| « Last Edit: Apr 22nd, 2007, 1:36pm by towr » |
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Grimbal
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Re: Longest Binary Number
« Reply #5 on: Apr 22nd, 2007, 2:52pm » |
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Would 10101 contain 2 times the sequence 101?
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Grimbal
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Re: Longest Binary Number
« Reply #6 on: Apr 22nd, 2007, 3:07pm » |
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I get 10001000101 or reversed.
Or 100010001010 depending on how you count the multiplicity.
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| « Last Edit: Apr 22nd, 2007, 3:10pm by Grimbal » |
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Eigenray
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Re: Longest Binary Number
« Reply #7 on: Apr 22nd, 2007, 4:29pm » |
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There are 6 permutations of 10100100001 that should also work.
Consider the lengths of the runs of 0s between adjacent 1s. There are no more than 3 runs.
If any has length 4, the others have length  2, and we can't have more than 2 of length  2. So if some run has length 4, the longest string has lengths (4,2,1).
Otherwise, there longest it could be is (3,3,1).
So there are a total of 9 strings with length 11, and none longer.
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| « Last Edit: Apr 22nd, 2007, 4:46pm by Eigenray » |
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rmsgrey
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Re: Longest Binary Number
« Reply #8 on: Apr 23rd, 2007, 7:18am » |
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on Apr 22nd, 2007, 1:27pm, towr wrote:| There are 8 three bit sequences, two contain 11 |
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technically, three contain 11 - 110, 011 and 111
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Grimbal
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Re: Longest Binary Number
« Reply #9 on: Apr 23rd, 2007, 8:21am » |
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Technically? As opposed to what?
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towr
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Re: Longest Binary Number
« Reply #10 on: Apr 23rd, 2007, 8:46am » |
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on Apr 23rd, 2007, 7:18am, rmsgrey wrote:| technically, three contain 11 - 110, 011 and 111 |
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Oops.. Yeah, that was careless of me to miss.
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rmsgrey
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Re: Longest Binary Number
« Reply #11 on: Apr 24th, 2007, 6:17am » |
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on Apr 23rd, 2007, 8:21am, Grimbal wrote:| Technically? As opposed to what? |
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As opposed to a wild estimate. Two, plus or minus one is reasonable compared to some of the estimates I've seen over the years...
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