The general any-number-base-b rule here (as others have noticed in base 10^k) is that 1/(b-1)^2 = 0.0123456... (where '1', '2'.. are base b digits). The origi

The general any-number-base-b rule here (as others have noticed in base 10^k) is that 1/(b-1)^2 = 0.0123456... (where '1', '2'.. are base b digits). The original post is this fact in base 1000. Proof for any base: 1/(b-1) = 1/b + 1/b^2 + 1/b^3 + .. = 0.11111.. (base b). So 1/(b-1)^2 = 0.11111.. * (1/b + 1/b^2 + 1/b^3 + ...) = 0.012345... QED. Richard Feynman beat us all to the punch here by noticing that 1/243 = 0.004115226337..., a fact which he wrote in a letter from a secret lab to someone in the outside world, and which put him under suspicion of sending secret messages! That gem of a fraction turns out to be a result of the above stuff as 1/243 = 111 * (1/999^2) + 4/999. Here's a slightly more detailed explanation: http://tylerneylon.com/b/archives/51 http://tylerneylon.com/b/archives/51