56 lines
2.2 KiB
Markdown
56 lines
2.2 KiB
Markdown
---
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id: 5900f5021000cf542c510015
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title: 'Problem 406: Guessing Game'
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challengeType: 5
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forumTopicId: 302074
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---
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# --description--
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We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions.
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Each number (question) we ask, we get one of three possible answers: "Your guess is lower than the hidden number" (and you incur a cost of a), or
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"Your guess is higher than the hidden number" (and you incur a cost of b), or
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"Yes, that's it!" (and the game ends).
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Given the value of n, a, and b, an optimal strategy minimizes the total cost for the worst possible case.
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For example, if n = 5, a = 2, and b = 3, then we may begin by asking "2" as our first question.
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If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that "1" is the hidden number (for a total cost of 3). If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be "4". If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that "3" is the hidden number (for a total cost of 2+3=5). If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that "5" is the hidden number (for a total cost of 2+2=4). Thus, the worst-case cost achieved by this strategy is 5. It can also be shown that this is the lowest worst-case cost that can be achieved. So, in fact, we have just described an optimal strategy for the given values of n, a, and b.
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Let C(n, a, b) be the worst-case cost achieved by an optimal strategy for the given values of n, a, and b.
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Here are a few examples: C(5, 2, 3) = 5 C(500, √2, √3) = 13.22073197... C(20000, 5, 7) = 82 C(2000000, √5, √7) = 49.63755955...
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Let Fk be the Fibonacci numbers: Fk = Fk-1 + Fk-2 with base cases F1 = F2 = 1.Find ∑1≤k≤30 C(1012, √k, √Fk), and give your answer rounded to 8 decimal places behind the decimal point.
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# --hints--
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`euler406()` should return 36813.12757207.
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```js
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assert.strictEqual(euler406(), 36813.12757207);
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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 euler406() {
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return true;
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}
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euler406();
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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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