63 lines
1.5 KiB
Markdown
63 lines
1.5 KiB
Markdown
---
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id: 5900f5141000cf542c510027
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title: 'Problem 423: Consecutive die throws'
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challengeType: 5
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forumTopicId: 302093
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dashedName: problem-423-consecutive-die-throws
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---
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# --description--
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Let $n$ be a positive integer.
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A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same value.
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For example, if $n = 7$ and the values of the die throws are (1, 1, 5, 6, 6, 6, 3), then the following pairs of consecutive throws give the same value:
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$$\begin{align}
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& (\underline{1}, \underline{1}, 5, 6, 6, 6, 3) \\\\
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& (1, 1, 5, \underline{6}, \underline{6}, 6, 3) \\\\
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& (1, 1, 5, 6, \underline{6}, \underline{6}, 3)
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\end{align}$$
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Therefore, $c = 3$ for (1, 1, 5, 6, 6, 6, 3).
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Define $C(n)$ as the number of outcomes of throwing a 6-sided die $n$ times such that $c$ does not exceed $π(n)$.<sup>1</sup>
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For example, $C(3) = 216$, $C(4) = 1290$, $C(11) = 361\\,912\\,500$ and $C(24) = 4\\,727\\,547\\,363\\,281\\,250\\,000$.
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Define $S(L)$ as $\sum C(n)$ for $1 ≤ n ≤ L$.
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For example, $S(50)\bmod 1\\,000\\,000\\,007 = 832\\,833\\,871$.
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Find $S(50\\,000\\,000)\bmod 1\\,000\\,000\\,007$.
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<sup>1</sup> $π$ denotes the prime-counting function, i.e. $π(n)$ is the number of primes $≤ n$.
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# --hints--
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`consecutiveDieThrows()` should return `653972374`.
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```js
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assert.strictEqual(consecutiveDieThrows(), 653972374);
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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 consecutiveDieThrows() {
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return true;
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}
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consecutiveDieThrows();
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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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