59 lines
2.0 KiB
Markdown
59 lines
2.0 KiB
Markdown
---
|
|
id: 5900f4ba1000cf542c50ffcd
|
|
title: 'Problem 334: Spilling the beans'
|
|
challengeType: 5
|
|
forumTopicId: 301992
|
|
dashedName: problem-334-spilling-the-beans
|
|
---
|
|
|
|
# --description--
|
|
|
|
In Plato's heaven, there exist an infinite number of bowls in a straight line. Each bowl either contains some or none of a finite number of beans. A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls. The game ends when each bowl contains either one or no beans.
|
|
|
|
For example, consider two adjacent bowls containing 2 and 3 beans respectively, all other bowls being empty. The following eight moves will finish the game:
|
|
|
|
<img class="img-responsive center-block" alt="animation of game when two adjacent bowls contains 2 and 3 beans respectivelly" src="https://cdn.freecodecamp.org/curriculum/project-euler/spilling-the-beans.gif" style="background-color: white; padding: 10px;">
|
|
|
|
You are given the following sequences:
|
|
|
|
$$\begin{align}
|
|
& t_0 = 123456, \\\\
|
|
& t_i = \begin{cases}
|
|
\frac{t_{i - 1}}{2}, & \text{if $t_{i - 1}$ is even} \\\\
|
|
\left\lfloor\frac{t_{i - 1}}{2}\right\rfloor \oplus 926252, & \text{if $t_{i - 1}$ is odd}
|
|
\end{cases} \\\\
|
|
& \qquad \text{where $⌊x⌋$ is the floor function and $\oplus$ is the bitwise XOR operator.} \\\\
|
|
& b_i = (t_i\bmod 2^{11}) + 1.
|
|
\end{align}$$
|
|
|
|
The first two terms of the last sequence are $b_1 = 289$ and $b_2 = 145$. If we start with $b_1$ and $b_2$ beans in two adjacent bowls, 3419100 moves would be required to finish the game.
|
|
|
|
Consider now 1500 adjacent bowls containing $b_1, b_2, \ldots, b_{1500}$ beans respectively, all other bowls being empty. Find how many moves it takes before the game ends.
|
|
|
|
# --hints--
|
|
|
|
`spillingTheBeans()` should return `150320021261690850`.
|
|
|
|
```js
|
|
assert.strictEqual(spillingTheBeans(), 150320021261690850);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function spillingTheBeans() {
|
|
|
|
return true;
|
|
}
|
|
|
|
spillingTheBeans();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|