| dyson numbers puzzle solution |
[May. 18th, 2009|01:37 am] |
What integer, expressed in decimal, has the property that, when you take its rightmost digit and move it to the front, it is exactly doubled in value?
We want to find a number X = d1d2d3⋯N (where di are the digits of the number when written in decimal and N is the number of digits) such that
2 × ( d1d2d3⋯dN-1dN) = dNd1d2d3⋯dN-1
which can be written in a more conventional notation as:
2 X = (X - dN)/10 + dN 10N-1
in which the "move the rightmost digit to the front" operation has been expressed in simple arithmetic.
Multiply by 10 and subtract X from both sides to get:
19 X = (10N - 1)dN
Because X is an integer, this tells us that 19 divides (10N-1)dN. Because 19 is prime, this tells us that either (10N-1) or dN (or both) must be divisible by 19. But dN is a single digit, so 0 < dN <= 10, and therefore not divisible by 19, so it must be the (10N-1) part that's divisible by 19. In modular arithmetic we can write this as:
10N ≡ 1 (mod 19)
By Fermat's Little Theorem, we see that one solution to this is N = 18. So our number has eighteen digits.
We now have:
19 X = (1018 - 1) dN
Apparently we get a solution for each possible choice of the final digit dN.
For dN=0 we get the trivial result X=0. Boring.
Choosing dN = 1 we get X = (1019-1)/19 = 52631578947368421. Twice this number is 105263157894736842... It works, but... there's a secret leading zero!
With dN = 2, we get X = 105263157894736842 and 2X = 210526315789473684, which works.
Apparently this puzzle got a bit of attention after its recent (off-hand) mention in the New York Times. Following that link you might find some easier/better solutions. This also (via co-worker Brian) clued me into the Tierney Lab column/blog (![[info]](http://l-stat.livejournal.com/img/syndicated.gif) tierneylab) which features puzzles on Mondays, and which just now coined the term "dyson number" for these sorts of numbers. It's a funny attribution since Mr. Dyson has nothing to do with the puzzle, but one must admit it sounds better than "parasitic number." |
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| puzzle |
[Apr. 26th, 2009|07:23 pm] |
via ![[info]](http://l-stat.livejournal.com/img/community.gif) mathematics:
What integer, expressed in decimal, has the property that, when you take its rightmost digit and move it to the front, it is exactly doubled in value? Corollary puzzles: tripled in value, quadrupled, etc...
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| puzzle |
[Feb. 24th, 2005|05:32 pm] |
This was posted on the ucberkeley community over the summer. I think it's a good puzzle: Two distinct numbers are selected from the set of integers from 3 to 98, inclusive. A woman is told the product of the numbers; a man is told their sum. The man and the woman have the following conversation:
Woman: I don't know what the numbers are.
Man: I know that already, but I don't know the numbers either.
Woman: In that case, I know what the numbers are.
Man: So do I.
What are the numbers? |
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