Re: [math-fun] 9*99 etc
Thanks, Neil! I don't know why I didn't think to check the OEIS. Jim, that is https://oeis.org/A027828.
Actually it's https://oeis.org/A027878.
If you find out something interesting, please add a comment there. There is a certain amount of info there already, of course.
I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there. Jim
The traffic jam of carries in the middle isn't too surprising -- I would be amazed if you could ever get anything sensible out of them. Similarly, the growing fixed zones at opposite ends of the number are not totally unexpected: the one on the left is due to the fact that Prod(1-10^(-k)) converges, while the one on the right is a similar phenomenon happening in the ring of 10-adic numbers. What made my jaw drop was the intermediate zone, between that growing fixed outer crust and the totally chaotic core. There is a "mantle" of stuff that clearly exhibits patterns, but the patterns shift slowly -- blocks of 0s growing by one at every step, for instance. And I don't understand why it exists at all. If we want to adopt Jim's "phase" metaphor, I understand the solid crust and the gaseous core, but not the liquid mantle. On Wed, Jul 13, 2016 at 11:15 AM, James Propp <jamespropp@gmail.com> wrote:
Thanks, Neil! I don't know why I didn't think to check the OEIS.
Jim, that is https://oeis.org/A027828.
Actually it's https://oeis.org/A027878.
If you find out something interesting, please add a comment there. There is a certain amount of info there already, of course.
I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Allan's reply caused me to remember that Wolfram's "A New Kind of Science" has a brief discussion of things like this. Maybe I can even find my copy at home. In fact, I also kind of recall having a conversation with Cris Moore and Stephen Wolfram in which Cris got Stephen to concede that some of the stuff related to not-quite-chaotic carrying is a counterexample to Wolfram's typology. (Wolfram's reaction was that of a physicist, not a mathematician; he felt with a principle that had only one or two exceptions was a pretty good principle.) Cris, do you remember more? Jim On Wednesday, July 13, 2016, Allan Wechsler <acwacw@gmail.com> wrote:
The traffic jam of carries in the middle isn't too surprising -- I would be amazed if you could ever get anything sensible out of them. Similarly, the growing fixed zones at opposite ends of the number are not totally unexpected: the one on the left is due to the fact that Prod(1-10^(-k)) converges, while the one on the right is a similar phenomenon happening in the ring of 10-adic numbers.
What made my jaw drop was the intermediate zone, between that growing fixed outer crust and the totally chaotic core. There is a "mantle" of stuff that clearly exhibits patterns, but the patterns shift slowly -- blocks of 0s growing by one at every step, for instance. And I don't understand why it exists at all. If we want to adopt Jim's "phase" metaphor, I understand the solid crust and the gaseous core, but not the liquid mantle.
On Wed, Jul 13, 2016 at 11:15 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Thanks, Neil! I don't know why I didn't think to check the OEIS.
Jim, that is https://oeis.org/A027828.
Actually it's https://oeis.org/A027878.
If you find out something interesting, please add a comment there. There is a certain amount of info there already, of course.
I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It was a long time ago :-O but I remember pointing out that carrying is kind of chaotic, but that computing the final result is nevertheless doable in polynomial time — in particular, that it’s not computationally universal. I was trying to convince Stephen that the distinction between “simple” and “universal” isn’t binary, but that computational complexity lets us think about lots of intermediate levels of complexity in between. - Cris
On Jul 13, 2016, at 12:16 PM, James Propp <jamespropp@gmail.com> wrote:
Allan's reply caused me to remember that Wolfram's "A New Kind of Science" has a brief discussion of things like this. Maybe I can even find my copy at home.
In fact, I also kind of recall having a conversation with Cris Moore and Stephen Wolfram in which Cris got Stephen to concede that some of the stuff related to not-quite-chaotic carrying is a counterexample to Wolfram's typology. (Wolfram's reaction was that of a physicist, not a mathematician; he felt with a principle that had only one or two exceptions was a pretty good principle.) Cris, do you remember more?
Jim
On Wednesday, July 13, 2016, Allan Wechsler <acwacw@gmail.com> wrote:
The traffic jam of carries in the middle isn't too surprising -- I would be amazed if you could ever get anything sensible out of them. Similarly, the growing fixed zones at opposite ends of the number are not totally unexpected: the one on the left is due to the fact that Prod(1-10^(-k)) converges, while the one on the right is a similar phenomenon happening in the ring of 10-adic numbers.
What made my jaw drop was the intermediate zone, between that growing fixed outer crust and the totally chaotic core. There is a "mantle" of stuff that clearly exhibits patterns, but the patterns shift slowly -- blocks of 0s growing by one at every step, for instance. And I don't understand why it exists at all. If we want to adopt Jim's "phase" metaphor, I understand the solid crust and the gaseous core, but not the liquid mantle.
On Wed, Jul 13, 2016 at 11:15 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Thanks, Neil! I don't know why I didn't think to check the OEIS.
Jim, that is https://oeis.org/A027828.
Actually it's https://oeis.org/A027878.
If you find out something interesting, please add a comment there. There is a certain amount of info there already, of course.
I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
participants (3)
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Allan Wechsler -
Cris Moore -
James Propp