Congratulations to this week's winner,
Carl Westine
A partial solution was submitted by Brian Bergdall. Correct solutions
came from Bryan Fluhrer, Philippe Fondanaiche, Sudipta Das, Francesc Suñol,
Alex Boisvert and Charles Rocca, Dane Brooke, Yves Thiry, Brian Laughlin,
Ivan Lisac.
First note that Yk + Nk =
10k - 1 . How much is Nk? In order
to have positive k digit number with no 9s, each of the digits must be
between 0 and 8, inclusive. There being 9 such digits to choose from,
we have Nk = 9k - 1, and so Yk
= 10k - 9k. Therefore Yk
/ Nk = (10k - 9k)/(9k
- 1) = ¥, by simple algebra, or by
complicated L'Hôpital's rule. Intutitively, this is saying
that as a number gets longer and longer, the chances of it not
having a 9 as a digit becomes increasingly negligible.
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