Re: [math-fun] Freeman Dyson integer problem
Let's try to "cycle": Is there an integer with digits abcdefg...klm where mabcdefg...kl is m times abcdefg...klm then lmabcdefg...kl is l times mabcdefg...kl then klmabcdefg...kl is k times mabcdefg...k then (etc.) Best, É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Cordwell, William R Envoyé : jeudi 26 mars 2009 18:57 À : math-fun Objet : Re: [math-fun] Freeman Dyson integer problem Using the same idea, start with 2, and consecutively divide by 3, to obtain 3*2068965517241379310345827586 = 6206896551724137931037482758 -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Thursday, March 26, 2009 11:16 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem For some reason I haven't quite figured out the algebra, but permit me to report that 923076/3 = 307692, with two digits moving to the back. On Thu, Mar 26, 2009 at 12:12 PM, Cordwell, William R <wrcordw@sandia.gov>wrote:
So, one can just start with a digit and start working backwards, at each point dividing by 2 and seeing if it "works" when the digit repeats? E.g., 315789473684210526 will also work; 421052631578947368, etc.
Bill C.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Veit Elser Sent: Thursday, March 26, 2009 9:17 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem
421052631578947368 / 210526315789473684 = 2
smallest k, such that 10^k - 2 is divisible by 2 x 10 - 1 = 19, is k = 17
Veit
On Mar 26, 2009, at 3:52 PM, Henry Baker wrote:
http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html
"At Jason, taking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it's possible to exactly double the value. Dyson will immediately say, "Oh, that's not difficult," allow two short beats to pass and then add, "but of course the smallest such number is 18 digits long." When this happened one day at lunch, William Press remembers, "the table fell silent; nobody had the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds." The meal then ended with men who tend to be described with words like "brilliant," "Nobel" and "MacArthur" quietly retreating to their offices to work out what Dyson just knew."
Is this correct?
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Same idea, dividing 4 into an initial digit of 1, next digit = 0, 4 into 10 = next digit = 2, 4 into 22 = next digit 5, 4 into 25 = next digit 6, 4 into 16 = 4, 4 into 4 = 1, repeat, so test, and 4*102564 = 410256 . One could envision Freeman Dyson doing this in his head for the relevant case of dividing by 2, although "2 seconds" is fairly sporty. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Eric Angelini Sent: Thursday, March 26, 2009 1:00 PM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem Let's try to "cycle": Is there an integer with digits abcdefg...klm where mabcdefg...kl is m times abcdefg...klm then lmabcdefg...kl is l times mabcdefg...kl then klmabcdefg...kl is k times mabcdefg...k then (etc.) Best, É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Cordwell, William R Envoyé : jeudi 26 mars 2009 18:57 À : math-fun Objet : Re: [math-fun] Freeman Dyson integer problem Using the same idea, start with 2, and consecutively divide by 3, to obtain 3*2068965517241379310345827586 = 6206896551724137931037482758 -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Thursday, March 26, 2009 11:16 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem For some reason I haven't quite figured out the algebra, but permit me to report that 923076/3 = 307692, with two digits moving to the back. On Thu, Mar 26, 2009 at 12:12 PM, Cordwell, William R <wrcordw@sandia.gov>wrote:
So, one can just start with a digit and start working backwards, at each point dividing by 2 and seeing if it "works" when the digit repeats? E.g., 315789473684210526 will also work; 421052631578947368, etc.
Bill C.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Veit Elser Sent: Thursday, March 26, 2009 9:17 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem
421052631578947368 / 210526315789473684 = 2
smallest k, such that 10^k - 2 is divisible by 2 x 10 - 1 = 19, is k = 17
Veit
On Mar 26, 2009, at 3:52 PM, Henry Baker wrote:
http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html
"At Jason, taking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it's possible to exactly double the value. Dyson will immediately say, "Oh, that's not difficult," allow two short beats to pass and then add, "but of course the smallest such number is 18 digits long." When this happened one day at lunch, William Press remembers, "the table fell silent; nobody had the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds." The meal then ended with men who tend to be described with words like "brilliant," "Nobel" and "MacArthur" quietly retreating to their offices to work out what Dyson just knew."
Is this correct?
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How about 11111111111? ;-) Actually, for mabcdef...jkl to be m times abc...klm, a would have to be 1, so there's no real way for your cycle to work unless all the values are 1. On Thu, Mar 26, 2009 at 12:00 PM, Eric Angelini <Eric.Angelini@kntv.be>wrote:
Let's try to "cycle":
Is there an integer with digits abcdefg...klm where mabcdefg...kl is m times abcdefg...klm then lmabcdefg...kl is l times mabcdefg...kl then klmabcdefg...kl is k times mabcdefg...k then (etc.)
Best, É.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] De la part de Cordwell, William R Envoyé : jeudi 26 mars 2009 18:57 À : math-fun Objet : Re: [math-fun] Freeman Dyson integer problem
Using the same idea, start with 2, and consecutively divide by 3, to obtain 3*2068965517241379310345827586 = 6206896551724137931037482758
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Thursday, March 26, 2009 11:16 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem
For some reason I haven't quite figured out the algebra, but permit me to report that 923076/3 = 307692, with two digits moving to the back.
On Thu, Mar 26, 2009 at 12:12 PM, Cordwell, William R <wrcordw@sandia.gov
wrote:
So, one can just start with a digit and start working backwards, at each point dividing by 2 and seeing if it "works" when the digit repeats? E.g., 315789473684210526 will also work; 421052631578947368, etc.
Bill C.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Veit Elser Sent: Thursday, March 26, 2009 9:17 AM To: math-fun Subject: Re: [math-fun] Freeman Dyson integer problem
421052631578947368 / 210526315789473684 = 2
smallest k, such that 10^k - 2 is divisible by 2 x 10 - 1 = 19, is k = 17
Veit
On Mar 26, 2009, at 3:52 PM, Henry Baker wrote:
http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html
"At Jason, taking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it's possible to exactly double the value. Dyson will immediately say, "Oh, that's not difficult," allow two short beats to pass and then add, "but of course the smallest such number is 18 digits long." When this happened one day at lunch, William Press remembers, "the table fell silent; nobody had the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds." The meal then ended with men who tend to be described with words like "brilliant," "Nobel" and "MacArthur" quietly retreating to their offices to work out what Dyson just knew."
Is this correct?
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participants (3)
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Cordwell, William R -
Dave Blackston -
Eric Angelini