## Ambiguous Numbers

### Problem 198

Published on Saturday, 14th June 2008, 02:00 am; Solved by 684; Difficulty rating: 80%A best approximation to a real number `x` for the denominator bound `d` is a rational number `r`/`s` (in reduced form) with `s` ≤ `d`, so that any rational number `p`/`q` which is closer to `x` than `r`/`s` has `q` > `d`.

Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6.
We shall call a real number `x` *ambiguous*, if there is at least one denominator bound for which `x` possesses two best approximations. Clearly, an ambiguous number is necessarily rational.

How many ambiguous numbers `x` = `p`/`q`,
0 < `x` < 1/100, are there whose denominator `q` does not exceed 10^{8}?