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Freeman Dyson's Problem

My friend John Lamperti turned my attention to a number theory problem in an article on Freeman Dyson in the March 29 New York Times Magazine Section. It is an excellent article, which I recommend. The section with the problem is the following:

[T]aking 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.


The discovery (or proof) of the smallest such number, 105263157894736842, makes a good problem for an elementary number theory course or a bright high school student.

The first time John told me the problem, he had heard it second-hand, and it was backwards: Is there an integer which if you take its first digit and move it to the back you can exactly double the value? In this case the answer is no, and the proof is simpler than the solution for Dyson’s problem.

To see my solutions, go here.

1 comment:

Dan Breslau said...

Wow, did this ever wake up my inner math geek :-)

I noticed some interesting properties in this solution:

o Of course, every digit N below the leading 1 is paired with either (2N mod 10), or ((2N+1) mod 10), to its left.

o The solution represents every digit twice, except for 0 and 9 which are represented once. It "feels" like this shouldn't be a coincidence, but I couldn't say why.

o 210526315789473684 and its double, 421052631578947368, can both be doubled in the same way.

o In fact, I'd wager that the number can be spliced at almost any digit, with the two halves swapped, yet still retain the same property -- *provided* that the splicing results in a leading digit that's less than 5. It even seems to hold if you allow 0 to be a leading digit: 2*052631578947368421 = 105263157894736842. (That gives us a lower solution to the original problem, so maybe we *shouldn't* allow 0 as a leading digit!)

Does a counterexample exist for this pattern?