66 lines
1.3 KiB
Markdown
66 lines
1.3 KiB
Markdown
---
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id: 5900f3a51000cf542c50feb8
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challengeType: 5
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title: 'Problem 57: Square root convergents'
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forumTopicId: 302168
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---
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## Description
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<section id='description'>
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It is possible to show that the square root of two can be expressed as an infinite continued fraction.
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√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...
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By expanding this for the first four iterations, we get:
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1 + 1/2 = 3/2 = 1.5
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1 + 1/(2 + 1/2) = 7/5 = 1.4
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1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
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1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...
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The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
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In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>euler57()</code> should return 153.
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testString: assert.strictEqual(euler57(), 153);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function euler57() {
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// Good luck!
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return true;
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}
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euler57();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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