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Topic: "Dot to dot" decimal expansions (Read 760 times) |
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Noke Lieu
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Hey guys. Been a while.
Just thought up something that I can't quite crack in hurry (actually, it's something that I figured someone here would know about, or then again, something no-one's considered before).
I was playing with decimal expansions... and I noticed that sometimes, you can connect the same digit pairs without crossing lines...
A trivial example would be (see potato picture)
12321/99999 does work; as does 12322/99999
123/999 doesn't...
I've been wondering about irrationals too:
Natch, pi doesn't work (fails at the first 2)
e does better, falling over at the first 9.
What's going on? Obviously, you should be able to construct an string for which this works, right? 
(It kinda reminds me of 4 Colour Theorem, reminds me of Langford Numbers, Turing machines too.)
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| « Last Edit: Sep 17th, 2014, 5:30pm by Noke Lieu » |
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Grimbal
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Re: "Dot to dot" decimal expansions
« Reply #1 on: Sep 18th, 2014, 7:26am » |
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Code: __ __ __
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0.123123123123...
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towr
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Re: "Dot to dot" decimal expansions
« Reply #2 on: Sep 18th, 2014, 10:43am » |
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Pi might very well work. If you have enough digits of pi, the first part should eventually be an even-length palindrome. And then you can get started on the next segment. (And there are plenty of other cases that might work.)
But if you're looking for irrationals, it's probably easiest to construct one that works.
like 0.010011100100100111001110011100100100111001...
or 0.110111001011101111000100110101011110011011110...
Any irrational with just two digits will work.
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| « Last Edit: Sep 18th, 2014, 10:44am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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Noke Lieu
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pen... paper... let's go! (and bit of plastic)
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Re: "Dot to dot" decimal expansions
« Reply #3 on: Sep 18th, 2014, 8:27pm » |
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Ah, well, you see, I was playing it with the rule of you had to pair them in the order they arrive.
Hence you'd have to pair 0.123123...
which in turn prohibits you connecting the 3s.
However, I really like the idea of carefully picking your pairing. Probably a more interesting activity!
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| « Last Edit: Sep 18th, 2014, 8:28pm by Noke Lieu » |
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