That's exactly the technique I was trying to apply for base 10. For some reason I slipped and solved 26S = 3a (where a is the leading digit) rather than 29S = 3a, so I wound up with a base-100 solution. On Thu, Mar 26, 2009 at 1:35 PM, Dave Blackston <hyperdex@gmail.com> wrote:
A few things about this.
My technique also works for base 100, which is what you are doing here.
X/100 + d/100 = 3X
X = d/299
for d=92, we get X=92/299=4/13, and since the order of 10 mod 13 is 6, we get a six digit answer.
307692 is the repeating block of the decimal expansion of 4/13.
For a smaller answer, try the repeating block of 2/13. 153846*3=461538.
On Thu, Mar 26, 2009 at 10:15 AM, Allan Wechsler <acwacw@gmail.com> wrote:
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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