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.


My kids just got a whoopee cushion and are now busy mastering the flatulent arts. It’s no longer safe to sit anywhere.


At my house we have a closet under the stairs. The door to it is in the back of another closet. It’s like Harry Potter’s room. It’s where my kids have secret meetings.

I crawl in there twice a year, once to get the Christmas stuff, and later to put it away again. But I’m not sure I’ll be able to find anything this year.

You see, I just caught a glimpse in there and the floor is covered with 6 inches of pure mess. The closet has become a black hole for junk, an archeological midden, the sewer bottom for missing toy pieces and socks and books. I know how King Augeas felt.

Hope there’s no pizza in there.

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