a2b2ef3f75
* fix: clean-up Project Euler 441-460 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
62 lines
2.2 KiB
Markdown
62 lines
2.2 KiB
Markdown
---
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id: 5900f52a1000cf542c51003b
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title: 'Problem 444: The Roundtable Lottery'
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challengeType: 5
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forumTopicId: 302116
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dashedName: problem-444-the-roundtable-lottery
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---
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# --description--
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A group of $p$ people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £$p$, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.
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An arbitrary person is chosen to be the first player. Going around the table, each player has only one of two options:
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1. The player can scratch his ticket and reveal its worth to everyone at the table.
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2. The player can trade his unscratched ticket for a previous player's scratched ticket, and then leave the game with that ticket. The previous player then scratches his newly-acquired ticket and reveals its worth to everyone at the table.
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The game ends once all tickets have been scratched. All players still remaining at the table must leave with their currently-held tickets.
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Assume that each player uses the optimal strategy for maximizing the expected value of his ticket winnings.
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Let $E(p)$ represent the expected number of players left at the table when the game ends in a game consisting of $p$ players (e.g. $E(111) = 5.2912$ when rounded to 5 significant digits).
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Let $S_1(N) = \displaystyle\sum_{p = 1}^N E(p)$.
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Let $S_k(N) = \displaystyle\sum_{p = 1}^N S_{k - 1}(p)$ for $k > 1$.
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Find $S_{20}({10}^{14})$ and write the answer as a string in scientific notation rounded to 10 significant digits. Use a lowercase `e` to separate mantissa and exponent. For example, the answer for $S_3(100)$ would be `5.983679014e5`.
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# --hints--
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`roundtableLottery()` should return a string.
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```js
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assert(typeof roundtableLottery() === 'string');
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```
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`roundtableLottery()` should return the string `1.200856722e263`.
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```js
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assert.strictEqual(roundtableLottery(), '1.200856722e263');
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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 roundtableLottery() {
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return true;
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}
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roundtableLottery();
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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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