67 lines
1.5 KiB
Markdown
67 lines
1.5 KiB
Markdown
|
---
|
||
|
|
id: 5900f54a1000cf542c51005c
|
||
|
|
title: 'Problem 477: Number Sequence Game'
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 302154
|
||
|
|
dashedName: problem-477-number-sequence-game
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
The number sequence game starts with a sequence S of N numbers written on a line.
|
||
|
|
|
||
|
|
Two players alternate turns. At his turn, a player must select and remove either the first or the last number remaining in the sequence.
|
||
|
|
|
||
|
|
The player score is the sum of all the numbers he has taken. Each player attempts to maximize his own sum.
|
||
|
|
|
||
|
|
If N = 4 and S = {1, 2, 10, 3}, then each player maximizes his score as follows:
|
||
|
|
|
||
|
|
Player 1: removes the first number (1)
|
||
|
|
|
||
|
|
Player 2: removes the last number from the remaining sequence (3)
|
||
|
|
|
||
|
|
Player 1: removes the last number from the remaining sequence (10)
|
||
|
|
|
||
|
|
Player 2: removes the remaining number (2)
|
||
|
|
|
||
|
|
Player 1 score is 1 + 10 = 11.
|
||
|
|
|
||
|
|
Let F(N) be the score of player 1 if both players follow the optimal strategy for the sequence S = {s1, s2, ..., sN} defined as:
|
||
|
|
|
||
|
|
s1 = 0
|
||
|
|
|
||
|
|
si+1 = (si2 + 45) modulo 1 000 000 007
|
||
|
|
|
||
|
|
The sequence begins with S = {0, 45, 2070, 4284945, 753524550, 478107844, 894218625, ...}.
|
||
|
|
|
||
|
|
You are given F(2) = 45, F(4) = 4284990, F(100) = 26365463243, F(104) = 2495838522951.
|
||
|
|
|
||
|
|
Find F(108).
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`euler477()` should return 25044905874565164.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.strictEqual(euler477(), 25044905874565164);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function euler477() {
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
euler477();
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
// solution required
|
||
|
|
```
|