49 lines
1.1 KiB
Markdown
49 lines
1.1 KiB
Markdown
|
---
|
||
|
|
id: 5900f51a1000cf542c51002d
|
||
|
|
title: 'Problem 430: Range flips'
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 302101
|
||
|
|
dashedName: problem-430-range-flips
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
N disks are placed in a row, indexed 1 to N from left to right.
|
||
|
|
|
||
|
|
Each disk has a black side and white side. Initially all disks show their white side.
|
||
|
|
|
||
|
|
At each turn, two, not necessarily distinct, integers A and B between 1 and N (inclusive) are chosen uniformly at random. All disks with an index from A to B (inclusive) are flipped.
|
||
|
|
|
||
|
|
The following example shows the case N = 8. At the first turn A = 5 and B = 2, and at the second turn A = 4 and B = 6.
|
||
|
|
|
||
|
|
Let E(N, M) be the expected number of disks that show their white side after M turns. We can verify that E(3, 1) = 10/9, E(3, 2) = 5/3, E(10, 4) ≈ 5.157 and E(100, 10) ≈ 51.893.
|
||
|
|
|
||
|
|
Find E(1010, 4000). Give your answer rounded to 2 decimal places behind the decimal point.
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`euler430()` should return 5000624921.38.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.strictEqual(euler430(), 5000624921.38);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function euler430() {
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
euler430();
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
// solution required
|
||
|
|
```
|