a9418a1fe9
* fix: clean-up Project Euler 221-240 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
73 lines
2.0 KiB
Markdown
73 lines
2.0 KiB
Markdown
---
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id: 5900f4531000cf542c50ff65
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title: 'Problem 230: Fibonacci Words'
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challengeType: 5
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forumTopicId: 301874
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dashedName: problem-230-fibonacci-words
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---
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# --description--
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For any two strings of digits, $A$ and $B$, we define $F_{A,B}$ to be the sequence ($A, B, AB, BAB, ABBAB, \ldots$) in which each term is the concatenation of the previous two.
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Further, we define $D_{A,B}(n)$ to be the $n^{\text{th}}$ digit in the first term of $F_{A,B}$ that contains at least $n$ digits.
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Example:
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Let $A = 1\\,415\\,926\\,535$, $B = 8\\,979\\,323\\,846$. We wish to find $D_{A,B}(35)$, say.
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The first few terms of $F_{A,B}$ are:
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$$\begin{align}
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& 1\\,415\\,926\\,535 \\\\
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& 8\\,979\\,323\\,846 \\\\
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& 14\\,159\\,265\\,358\\,979\\,323\\,846 \\\\
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& 897\\,932\\,384\\,614\\,159\\,265\\,358\\,979\\,323\\,846 \\\\
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& 14\\,159\\,265\\,358\\,979\\,323\\,846\\,897\\,932\\,384\\,614\\,15\color{red}{9}\\,265\\,358\\,979\\,323\\,846
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\end{align}$$
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Then $D_{A,B}(35)$ is the ${35}^{\text{th}}$ digit in the fifth term, which is 9.
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Now we use for $A$ the first 100 digits of $π$ behind the decimal point:
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$$\begin{align}
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& 14\\,159\\,265\\,358\\,979\\,323\\,846\\,264\\,338\\,327\\,950\\,288\\,419\\,716\\,939\\,937\\,510 \\\\
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& 58\\,209\\,749\\,445\\,923\\,078\\,164\\,062\\,862\\,089\\,986\\,280\\,348\\,253\\,421\\,170\\,679
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\end{align}$$
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and for $B$ the next hundred digits:
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$$\begin{align}
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& 82\\,148\\,086\\,513\\,282\\,306\\,647\\,093\\,844\\,609\\,550\\,582\\,231\\,725\\,359\\,408\\,128 \\\\
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& 48\\,111\\,745\\,028\\,410\\,270\\,193\\,852\\,110\\,555\\,964\\,462\\,294\\,895\\,493\\,038\\,196
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\end{align}$$
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Find $\sum_{n = 0, 1, \ldots, 17} {10}^n × D_{A,B}((127 + 19n) × 7^n)$.
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# --hints--
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`fibonacciWords()` should return `850481152593119200`.
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```js
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assert.strictEqual(fibonacciWords(), 850481152593119200);
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```
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# --seed--
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## --seed-contents--
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```js
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function fibonacciWords() {
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return true;
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}
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fibonacciWords();
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```
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# --solutions--
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```js
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// solution required
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```
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