Freeman Dyson and 1/19

I just read this article about Freeman Dyson and a math puzzle that asks you to find a number that is doubled when you tear off the rightmost digit and stick it on the left. For example, tearing the 2 off 12 and sticking it in front gets you 21, which isn’t 2×12, so 12 isn’t right. Moving the 1 in 7654321 gets you 1765432 which also dosn’t double it, so that’s not the number either.

Anyway, it turns out the number we’re looking for is 18 digits long. It’s 052,631,578,947,368,421 which doubles to 105,263,157,894,736,842. (You’re allowed to stick a zero in front.)

Well, my son loves this kind of base-10 number tomfoolery, so I asked him if he could figure it out. And he reeled off the answer without pause. All 18 digits.

“Uhh, how’d ya know?” I asked.

He said it’s because 1/19 is 0.052631578947368421… (repeated forever), and 2/19 is 0.105263157894736842… (just move the 1). And of course 2/19 is twice 1/19. So if you know your N/19 repeating decimals, this is apparently easy peasy. And all the N/19′s are buried in that infinite sequence. You just have to start at different places. 3/19 is 0.157894736842105263…, 4/19 is 0.210526315789473684…, 5/19 is 0.263157894736842105…, etc.

So the next time someone asks “what’s a number that’s quadrupled when you tear the last two digits off the right and stick them on the left” you can say “doy, it’s 052,631,578,947,368,421 of course” and then roll your eyes. Because you’ve seen 1/19 and 4/19 written out as decimals.

Anyway, my son also told me there’s something similar for 1/29, except you triple the number by tearing off the rightmost digit and putting it on the left. I think it’s 28 digits long.

And also for 1/39, except you quadruple (x4) the number. And with 1/49 you x5 the number, and 1/59 gets you x6. Even 1/99 (0.0101010101…) works (x10 gives you 0.1010101010…).

As for 1/109, I don’t know, I’ll bet there’s some kind of x11 trick there. My son’s in bed now, but I’ll ask him tomorrow when he gets up.

Comments

60 Responses to “Freeman Dyson and 1/19”

  1. Neal on April 11th, 2009 10:24 am

    One other note: The linked NYTimes article describes how to generate 052,631,578,947,368,421. Write down a 1, double it and put down the 2, double the 2 and put down the 4, and keep going until you cycle all the way back. The sequence looks like this:

    1
    21
    421
    8421
    (16)8421

    When you get to the 16 you just put down the 6 and carry the 1. Then you double the 6 and add the carry. You get:

    (16)8421 -> (+1)68421 -> (6+6+1)68421 -> (13)68421 -> (+1)368421

    At the next step you get

    (3+3+1)368421 -> 7368421

    Eventually you end up back at 1 (with no carry) and the whole thing starts again.

  2. Neal on April 11th, 2009 11:14 am

    Another note: The method described in my comment above is the same as shifting and summing summing all the powers of 2. Like this:

    xxxxxxxxxxxxxxxxxxxxxxxxxx1
    xxxxxxxxxxxxxxxxxxxxxxxxx2
    xxxxxxxxxxxxxxxxxxxxxxxx4
    xxxxxxxxxxxxxxxxxxxxxxx8
    xxxxxxxxxxxxxxxxxxxxx16
    xxxxxxxxxxxxxxxxxxxx32
    xxxxxxxxxxxxxxxxxxx64
    xxxxxxxxxxxxxxxxx128
    xxxxxxxxxxxxxxxx256
    xxxxxxxxxxxxxxx512
    xxxxxxxxxxxxx1024
    xxxxxxxxxxxx2048
    xxxxxxxxxxx4096
    xxxxxxxxxx8192
    xxxxxxxx16384
    xxxxxxx32768
    xxxxxx65536
    xxxx131072
    xxx262144
    xx524288
    1048576
    zzzzzz
    ---------------------------
    zzzzzz421052631578947368421
    

    If you keep doing this forever, with all the powers of 2 (to infinity), you end up with …052631578947368421 repeated over and over. Since you repeat on the left and the sequence is infinite, the first digit is undefined. If you start with 0.XXXX and glue the infinite sequence after the decimal point, there are 18 places in the sequence where you could start. They correspond to the decimal representations of 1/19 thru 18/19.

  3. Neal on April 12th, 2009 7:35 pm

    There is another way to generate this sequence, but left-to-right instead of right-to-left like we’ve done so far. It uses powers of 5 instead of powers of 2, and it shifts 2 by digits (not just 1) for each new term. Here’s how it looks:

    1
    x05
    xxx25
    xxxx125
    xxxxxx625
    xxxxxxx3125
    xxxxxxxx15625
    xxxxxxxxxx78125
    xxxxxxxxxxx390625
    xxxxxxxxxxxx1953125
    xxxxxxxxxxxxxx9765625
    xxxxxxxxxxxxxxx48828125
    xxxxxxxxxxxxxxxx244140625
    xxxxxxxxxxxxxxxxx1220703125
    xxxxxxxxxxxxxxxxxxx6103515625
    xxxxxxxxxxxxxxxxxxxx30517578125
    -------------------------------
    10526315789473684210xxxxxxxxxxx
    

    You can also do this 2 digits at a time but you have to predict how much carrying you’re going to get. It’s not hard though.

    And if you do generate this 2 digits at a time, you can start anywhere. You can start with 10 or 05 or 52 or 26. You don’t have to worry about even or odd alignment — it works both ways.

    Since you can generate the 1/19 sequence right-to-left with powers of 2, or left-to-right with powers of 5, what about 1/49? The post mentions that you can generate the 1/49 sequence (42 digits before it repeats) right-to-left with powers of 5. So can you also generate 1/49 left-to-right with powers of 2?

    Yes. 1/49 is 0.020408163265306122448979591836734693877551… You can travel left-to-right through this series 2 digits at a time by doubling the previous 2 digits mod 100. And if the result is 50 or above you add one (to carry from the next double). So from 02 you get 04, 08, 16, 32, and then 65 (you add 1 to 64 because it 64 >= 50). From 65 you get 130 which is 30 when you mod 100.

    What’s really cool about this sequence is that you can start doubling from anywhere, even though this generates 2 digits at a time. You can start with 02 or with 20. Or with 08 or 81 or 16. It’s like we have two sequences of 2-digit numbers that zip together. The even-aligned sequence (02 04 08 16 32 65 30 etc) and the odd-aligned sequence (20 40 81 63 26 53 06 12 24 etc). Both follow the same doubling rule, and both are the same when viewed as a sequence of 1-digit numbers.

  4. SoldAtTheTop on April 13th, 2009 9:17 pm

    WOW! That is really cool… How do you think he figured that n/19 pattern?

    Did he already know the pattern or did he figure a clever way to solve the problem and then seek the pattern out?

  5. Neal on April 13th, 2009 11:50 pm

    Hey SATT, nice to see you!

    My son likes calculating stuff out and looking for patterns. He’s done it since he was 2. Squares, cubes, primes, Pascal triangle stuff, all kinds of integer sequences.

    And you know how N/7 works:

    1/7 0.142857…
    2/7 0.285714…
    3/7 0.428571…
    4/7 0.571428…
    5/7 0.714285…
    9/7 0.857142…

    Same 6 digits (repeated forever), just started at different places. And 14*2=28, 28*2=56, and you add to 56 to make 57 because 57*2=114 which is 3 digits so you have to take the 1 off and carry it back to the 56. Or you can start doubling at 42 instead of 14 — the even/odd patterns zip together.

    So once my son saw how the 7ths worked he’s looked for the same thing elsewhere. The 19ths are almost the same.

  6. SoldAtTheTop on April 14th, 2009 11:30 am

    WOW AGAIN!… Ill have to start taking a closer look at numbers… I never realized how interesting all these natural patterns are.

  7. Eric Binnendyk on November 4th, 2010 5:55 pm

    for 1/109, you ADD ONE to the decimal after putting the last digit at the front. That multiplies it by eleven.

  8. view site on July 7th, 2012 8:58 am

    i such as chaning tatum because he has a great body, just
    look at those chest muscles.

  9. scholarships for college students|scholarships for college students in texas|2011 scholarships for college students|scholarships for college students in california|scholarships for college students 2011|easy scholarships for college students|grants and ss on July 8th, 2012 2:28 pm

    I also conceive thence , perfectly indited post!
    .

  10. grants for single mothers in florida on July 8th, 2012 3:38 pm

    Spot up for this writeup, I honestly think this
    fabulous website wants a large amount more consideration.
    Ill likely to end up again you just read a great deal more, thanks for the info.

  11. financial assistance for single mothers in florida on July 8th, 2012 4:29 pm

    It¦s really a great and useful piece of information. I¦m happy someone to only shared this useful info with us.
    Please stay us informed along the lines of this. Thanks for sharing.

  12. cell phone lookup on July 8th, 2012 4:47 pm

    every woman needs good jewelries because it becomes a fashion statement of your personality

  13. free scholarships for single mothers on July 8th, 2012 5:08 pm

    In my opinion ,, truth be told, that the achievements work with respect
    to basically the% excellent results is% response to these letdown.

  14. cell phone lookup for free on July 10th, 2012 4:27 am

    A individual essentially help to make significantly
    articles I would state. This is that the very first time I frequented your website page and so
    far I surprised with the research you made to create this particular
    publish incredible. Wonderful task!

  15. scholarships for women on July 11th, 2012 4:34 am

    I only added this site to my bookmarks. I love reading you.
    It had been quite helpful and solved that the issue tons

  16. http://www.prlog.org/11734629-phone-number-search-reverse-number-lookup.html on July 16th, 2012 1:09 pm

    That is really fascinating, You are an excessively professional
    blogger. I’ve joined your rss feed and stay up for searching for more of your magnificent post. Additionally, I’ve shared your site in my social networks

  17. cell phone number lookup on July 16th, 2012 2:15 pm

    I and also my guys came digesting the fine tips and
    tricks found on that the website then speedily I got a terrible feeling
    I never thanked that the blog owner for those techniques.
    Most of the ladies are really totally thrilled to read through them and have now sincerely
    been loving those things. Thanks for getting very considerate
    and for picking varieties of marvelous subject matter most people have been really eager to obtain.
    My very own honest apologies for not expressing gratitude to you sooner.

  18. reverse phone on July 16th, 2012 2:30 pm

    my cd compilation have grown up to be very very large, i
    love listening to music all day long

  19. http://www.prlog.org/11734648-criminal-background-check-public-records.html on July 16th, 2012 3:29 pm

    After I originally commented I clicked the Notify me when new comments are added checkbox and now every time a comment
    is added I learn four emails with the identical comment.
    Is there any line of attack youll have the capacity to remove me out of that service Thanks!

  20. http://www.prlog.org/11674605-reverse-phone-lookup-finders.html on July 16th, 2012 3:39 pm

    Wonderful beat ! I wish to apprentice while you amend your site, how can i subscribe for a blog
    website? The account helped me a acceptable deal.

    I had been a little bit acquainted of this your
    broadcast offered bright clear idea

  21. phone number lookup on July 16th, 2012 6:16 pm

    my daughter are doing some scrapbooking stuffs
    over the years and she actually likes it,.

  22. http://www.prlog.org/11734625-reverse-cell-phone-directory-finds-anyone-with-telephone-number.html on July 16th, 2012 8:30 pm

    Thanks for finally talking about > Freeman Dyson and 1/19 : Code Obscurata < Loved it!

  23. http://ezinearticles.com/?Reverse-Address-Lookup---Get-the-Information-You-Need-the-Easy-Way&id=5909554 on July 16th, 2012 8:58 pm

    Spot up with this writeup, I honestly think this web site requirements a great deal more consideration.
    Ill oftimes be once more to discover a great deal more, thank you that information.

  24. http://officialreversephonelookup.com/ on July 16th, 2012 9:17 pm

    Hello, Neat post. There is a problem with your site in web explorer, may check this?
    IE nonetheless is the marketplace leader
    and a good section of other folks will omit your
    excellent writing due to this problem.

  25. cell phone lookup on July 16th, 2012 10:24 pm

    Ahaa, its pleasant discussion on the topic of this post at this place
    at this webpage, I have read all that, so now me also commenting at this
    place.

  26. phone number search on July 17th, 2012 2:29 am

    i love funny stuffs, then again i specially for instance funny movies and funny videos on the web

  27. people search on July 17th, 2012 2:39 am

    Can I merely say thats a relief to search for somebody that actually knows what theyre discussing
    on that the net. You certainly discover how to bring an issue to light and make it critical.
    The diet have to check this out and understand this side of
    your story. I cant believe youre less wide-ranging when you certainly develop that the gift.

  28. reverse phone number lookup on July 17th, 2012 2:46 am

    Im impressed, I must say. Really rarely can i encounter a weblog thats both educative and
    entertaining, and without a doubt, you can have hit the nail for the
    head. Your notion is outstanding the pain is an issue that not enough people are speaking intelligently about.
    Were very happy i always found this during my get a hold of
    some thing concerning this.

  29. reverse phone lookup on July 18th, 2012 1:45 am

    Researchers have prepared an animal model of major depressive disorder that is certainly depending
    on an unusual genetic mutation that appears to cause the condition in most of people who inherit it

  30. http://www.prlog.org/11261550-phone-number-lookup-catch-cheater-quickly.html on July 25th, 2012 2:06 am

    Im not from a position to view this website correctly
    on saffari I think theres a drawback

  31. ways to make money on July 27th, 2012 6:43 am

    I concur with your conclusions and will eagerly watch forward to your future updates.

    That the usefulness and significance is overwhelming and has been invaluable to me!

  32. reverse phone on August 1st, 2012 10:14 am

    I am curious to lookup out out what blog platform you are working with Im experiencing
    some little security issues with my latest site and
    I would just like to search for something more riskfree.
    Do you have any other suggestions

  33. farmville cheats on August 11th, 2012 2:43 pm

    Preceding to you decide to create your own checklist to include an
    idea associated with what camping checklist ought to. Actually A listing
    can be more than what you will need.

  34. cell phone lookup on August 11th, 2012 2:55 pm

    we are using plastic kitchen faucets at home as that they are very inexpensive and it is promising to with
    no trouble replace them if that they broke

  35. phone number search app on August 11th, 2012 3:06 pm

    It’s always so nice and also packed with amusement for me and my office acquaintances to visit your site at the very least thrice weekly to read through the newest guidance you have.

  36. cell phone lookup on August 11th, 2012 3:07 pm

    This web site is mostly a walkthrough it really is the internet
    it suited you with this and didnt be acquainted with who to question.

    Glimpse here, and youll undoubtedly discover it.

  37. cell phone lookup on August 11th, 2012 3:14 pm

    Great V I should certainly pronounce, impressed with
    your website. I had no trouble navigating through all
    tabs as well as related info ended up being truly simple to do to
    access. I recently found what I hoped for before you identify it in the
    least. Reasonably unusual. Is likely to appreciate it for
    those who add forums or anything, web site theme . a tones system for
    the client to communicate. Fine task.

  38. reverse cell phone lookup on August 11th, 2012 6:24 pm

    An impressive share, I simply with all this onto a colleague who was merely carrying
    out a little analysis for this. And that he in truth bought me breakfast since
    I ran across it for him. smile. So i want to reword that Thnx for
    that the treat! However yeah Thnkx for spending some time to talk about this, I feel strongly
    concerning this and love reading regarding this topic.
    Whenever probable, as you become expertise, do you mind updating your blog post with an increase
    of details It truly is highly a first-class choice for
    me. Big thumb up due to this post!

  39. reverse cell phone lookup on August 11th, 2012 6:44 pm

    I see someone to have been using WordPress on your blog, wordpress is
    the best.

  40. cell phone lookup on August 11th, 2012 6:46 pm

    Spot ill carry on with this writeup, I seriously feel this fabulous website wants far more consideration.
    Ill apt to be again to study much more, appreciate your that information.

  41. reverse cell phone lookup on August 11th, 2012 6:53 pm

    Ive been exploring for a small for any other highquality articles or blog posts on
    this sort of area . Exploring in Yahoo I ultimately stumbled upon this website.
    Studying this info So im satisfied to convey that Ive a very first-class uncanny feeling I found out exactly what I needed.

    I most unquestionably will make certain to dont omit this website and provides it a glance
    a continuing.

  42. individual health insurance on August 12th, 2012 8:18 am

    Would love to perpetually get updated outstanding
    website ! .

  43. free online reverse phone lookup with name on August 12th, 2012 10:43 am

    You have a very nice layout for your blog, i want it to utilize on my site too .

  44. reverse cell phone lookup on August 12th, 2012 10:43 am

    Awesome post, will be a daily visitor out of now on!

  45. free reverse phone lookup google usa on August 12th, 2012 10:46 am

    appreciation for writing!. “It’s as that they took the straightforward approach out, That rivers and people, go crooked. by Jill Peterson.

  46. free reverse cell phone lookup reviews on August 12th, 2012 10:48 am

    Howdy, Could I export your page picture and utilize
    it on my blog page

  47. cell phone number search by number free on August 12th, 2012 11:00 am

    First-rate post. I was checking continuously this blog and I am impressed.

  48. pharmacist on August 12th, 2012 11:21 am

    Hello! I would for instance to give out a enormous thumbs up for your
    fantastic information youve got here during this post.

    I will be returning to your blog to locate more soon.

  49. reverse phone on August 12th, 2012 12:25 pm

    Hey, I lately came to this internet site and Ive visited it daily because that so that you can still remain existing.

    Maintain up the great function.

  50. real estate on August 12th, 2012 2:07 pm

    This blog is definitely rather handy since Im at the moment creating an internet floral website although I am
    merely starting out therefore its actually fairly small, nothing along the lines of this site.
    Can link to a few of the posts here as that they are quite.
    Thanks much. Zoey Olsen

  51. reverse phone lookup on August 12th, 2012 2:19 pm

    Considerably, the post is actually the greatest on this worthy topic.

    I agree with your conclusions and can eagerly look forward to your future updates.
    Simply just saying thanks can not simply be sufficient, for that the fantasti
    c clarity in your writing. I will perfect away grab your rss feed to stay abreast of any other updates.
    Genuine work and also much success in your
    business dealings!

  52. address lookup on August 12th, 2012 2:23 pm

    You have been so cool man, the post on your domains have been super great.
    ,

  53. free ringtones on August 12th, 2012 4:16 pm

    Hiya, have you by chance pondered to create concerning Nintendo or PS handheld

  54. reverse phone lookup on August 12th, 2012 4:45 pm

    Condense! I believe it will be a lot more efficient if
    presented better.

  55. reverse cell phone lookup on August 12th, 2012 5:25 pm

    there have been tanning lotions which are very demanding on the
    skin, i always use organic tanning lotions,,

  56. free cell phone lookup name canada on August 12th, 2012 5:33 pm

    There are a couple of fascinating points over
    time in the following paragraphs on the other
    hand I dont learn if I see them all center to heart. There is
    certainly some validity however I most certainly will take hold opinion until I look into it further.

    Fine post , thanks therefore we want much
    more! Added onto FeedBurner in addition

  57. free online reverse phone number lookup on August 12th, 2012 5:43 pm

    An interesting discussion may be worth comment. I do think you to
    only write on this topic, it will not certainly be
    a taboo subject in spite of this usually consumers are insufficient to communicate on such topics.
    To a higher. Cheers

  58. free reverse phone lookup google with name on August 12th, 2012 5:53 pm

    I think your blog is getting more and much more visitors.

  59. free dating sites on August 12th, 2012 6:20 pm

    when i do get my retirement, i would actually love to relax near the beach and enjoy a home on a tropical country,.

  60. people search on August 12th, 2012 10:30 pm

    I actually appreciate this post. Ive been seeking all over
    for this! Thank God I found it on Bing. Youve
    made my day! Thx again