Freeman Dyson Puzzle Solution

The puzzle from Freeman Dyson is Very Smart is:

  • Is (there) 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.”

My reasoning toward an answer goes like this:

The number should end in 2 (and start with 1) to make it the smallest possible such number. Then the number and its double must look like this:

  1nn...nn2
x         2
-----------
  21nn...nn

2 x 2 is 4, so:
  1nn...nn2
x         2
-----------
  21nn...n4

That means the second-to-last digit has to be 4, so:

  1nn...n42
x         2
-----------
  21nn...n4

2 x 4 is 8, so:

  1nn...n42
x         2
-----------
  21nn...84

That means the third-to-last digit is 8, so:

  1nn...842
x         2
-----------
  21nn...84

2 x 8 is 16, so:

        1
  1nn....842
x          2
------------
  21nn...684

That means the fourth-to-last digit is 6, so:
        1
  1nn...6842
x          2
------------
  21nn...684

Now it changes because of the carried 1. 2 x 6 + 1 is 13, so:
        11
  1nn....6842
x           2
-------------
  21nn...3684
Following on like that ends up with:

   1 1  1111 1 11
  105263157894736842
x                  2
--------------------
  210526315789473684

So the answer is 105263157894736842. That’s the smallest number that works for this puzzle, but 210526315789473684 is not the smallest number where the reverse can be done: what is the smallest integer where you can move the first (most significant) digit to the right, and the result will be exactly half the original number?

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One thought on “Freeman Dyson Puzzle Solution

  1. Pingback: Geoff Canyon’s Appeal to Authority

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