5a52c229f5
* fix: clean-up Project Euler 181-200 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing delimiter Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
43 lines
1.2 KiB
Markdown
43 lines
1.2 KiB
Markdown
---
|
|
id: 5900f4331000cf542c50ff45
|
|
title: 'Problem 198: Ambiguous Numbers'
|
|
challengeType: 5
|
|
forumTopicId: 301836
|
|
dashedName: problem-198-ambiguous-numbers
|
|
---
|
|
|
|
# --description--
|
|
|
|
A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac{r}{s}$ (in reduced form) with $s ≤ d$, so that any rational number $\frac{p}{q}$ which is closer to $x$ than $\frac{r}{s}$ 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. $\frac{9}{40}$ has the two best approximations $\frac{1}{4}$ and $\frac{1}{5}$ 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 $x = \frac{p}{q}$, $0 < x < \frac{1}{100}$, are there whose denominator $q$ does not exceed ${10}^8$?
|
|
|
|
# --hints--
|
|
|
|
`ambiguousNumbers()` should return `52374425`.
|
|
|
|
```js
|
|
assert.strictEqual(ambiguousNumbers(), 52374425);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function ambiguousNumbers() {
|
|
|
|
return true;
|
|
}
|
|
|
|
ambiguousNumbers();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|