71 lines
2.3 KiB
Markdown
71 lines
2.3 KiB
Markdown
---
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id: 5900f5191000cf542c51002b
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challengeType: 5
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title: 'Problem 428: Necklace of Circles'
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forumTopicId: 302098
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---
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## Description
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<section id='description'>
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Let <var>a</var>, <var>b</var> and <var>c</var> be positive numbers.
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Let W, X, Y, Z be four collinear points where |WX| = <var>a</var>, |XY| = <var>b</var>, |YZ| = <var>c</var> and |WZ| = <var>a</var> + <var>b</var> + <var>c</var>.
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Let C<sub>in</sub> be the circle having the diameter XY.
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Let C<sub>out</sub> be the circle having the diameter WZ.
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The triplet (<var>a</var>, <var>b</var>, <var>c</var>) is called a <em>necklace triplet</em> if you can place <var>k</var> ≥ 3 distinct circles C<sub>1</sub>, C<sub>2</sub>, ..., C<sub><var>k</var></sub> such that:
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<ul><li>C<sub><var>i</var></sub> has no common interior points with any C<sub><var>j</var></sub> for 1 ≤ <var>i</var>, <var>j</var> ≤ <var>k</var> and <var>i</var> ≠ <var>j</var>,</li><li>C<sub><var>i</var></sub> is tangent to both C<sub>in</sub> and C<sub>out</sub> for 1 ≤ <var>i</var> ≤ <var>k</var>,</li><li>C<sub><var>i</var></sub> is tangent to C<sub><var>i</var>+1</sub> for 1 ≤ <var>i</var> < <var>k</var>, and</li><li>C<sub><var>k</var></sub> is tangent to C<sub>1</sub>.</li></ul>
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For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.
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<img src="https://projecteuler.net/project/images/p428_necklace.png" alt="a visual reresentation of a necklace triplet">
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Let T(<var>n</var>) be the number of necklace triplets (<var>a</var>, <var>b</var>, <var>c</var>) such that <var>a</var>, <var>b</var> and <var>c</var> are positive integers, and <var>b</var> ≤ <var>n</var>.
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For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.
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Find T(1 000 000 000).
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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>necklace(1000000000)</code> should return 747215561862.
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testString: assert.strictEqual(necklace(1000000000), 747215561862);
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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 necklace(n) {
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// Good luck!
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return true;
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}
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necklace(1000000000)
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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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