freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-391-hopping-game.md

---
id: 5900f4f31000cf542c510006
title: 'Problem 391: Hopping Game'
challengeType: 5
forumTopicId: 302056
dashedName: problem-391-hopping-game
---

# --description--

Let sk be the number of 1’s when writing the numbers from 0 to k in binary.

For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so s5 = 7.

The sequence S = {sk : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.

A game is played by two players. Before the game starts, a number n is chosen. A counter c starts at 0. At each turn, the player chooses a number from 1 to n (inclusive) and increases c by that number. The resulting value of c must be a member of S. If there are no more valid moves, the player loses.

For example: Let n = 5. c starts at 0. Player 1 chooses 4, so c becomes 0 + 4 = 4. Player 2 chooses 5, so c becomes 4 + 5 = 9. Player 1 chooses 3, so c becomes 9 + 3 = 12. etc. Note that c must always belong to S, and each player can increase c by at most n.

Let M(n) be the highest number the first player can choose at her first turn to force a win, and M(n) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.

Given Σ(M(n))3 = 8150 for 1 ≤ n ≤ 20.

Find Σ(M(n))3 for 1 ≤ n ≤ 1000.

# --hints--

`euler391()` should return 61029882288.

```js
assert.strictEqual(euler391(), 61029882288);
```

# --seed--

## --seed-contents--

```js
function euler391() {

  return true;
}

euler391();
```

# --solutions--

```js
// solution required
```
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								---
 								id: 5900f4f31000cf542c510006
 								title: 'Problem 391: Hopping Game'
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								challengeType: 5
-												fix(curriculum): Added forumTopicId to remaining 1200 challeng… (#36558)


											
										
										
											2019-08-05 16:17:33 +00:00
+								forumTopicId: 302056
-												feat(curriculum): restore seed + solution to Chinese (#40683)

* feat(tools): add seed/solution restore script

* chore(curriculum): remove empty sections' markers

* chore(curriculum): add seed + solution to Chinese

* chore: remove old formatter

* fix: update getChallenges

parse translated challenges separately, without reference to the source

* chore(curriculum): add dashedName to English

* chore(curriculum): add dashedName to Chinese

* refactor: remove unused challenge property 'name'

* fix: relax dashedName requirement

* fix: stray tag

Remove stray `pre` tag from challenge file.

Signed-off-by: nhcarrigan <nhcarrigan@gmail.com>

Co-authored-by: nhcarrigan <nhcarrigan@gmail.com>
											
										
										
											2021-01-13 02:31:00 +00:00
+								dashedName: problem-391-hopping-game
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								---
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								# --description--
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								Let sk be the number of 1’s when writing the numbers from 0 to k in binary.
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so s5 = 7.
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								The sequence S = {sk : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
 								A game is played by two players. Before the game starts, a number n is chosen. A counter c starts at 0. At each turn, the player chooses a number from 1 to n (inclusive) and increases c by that number. The resulting value of c must be a member of S. If there are no more valid moves, the player loses.
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								For example: Let n = 5. c starts at 0. Player 1 chooses 4, so c becomes 0 + 4 = 4. Player 2 chooses 5, so c becomes 4 + 5 = 9. Player 1 chooses 3, so c becomes 9 + 3 = 12. etc. Note that c must always belong to S, and each player can increase c by at most n.
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
 								Let M(n) be the highest number the first player can choose at her first turn to force a win, and M(n) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.
 								Given Σ(M(n))3 = 8150 for 1 ≤ n ≤ 20.
 								Find Σ(M(n))3 for 1 ≤ n ≤ 1000.
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								# --hints--
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								`euler391()` should return 61029882288.
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								```js
 								assert.strictEqual(euler391(), 61029882288);
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								```
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								# --seed--
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								## --seed-contents--
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
 								```js
 								function euler391() {
-												fix: remove Good luck! comment from challenge seed code (#39575)


											
										
										
											2020-09-15 16:57:40 +00:00
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
+								  return true;
 								}
 								euler391();
 								```
-												Feat: add new Markdown parser (#39800)

and change all the challenges to new `md` format.
											
										
										
											2020-11-27 18:02:05 +00:00
+								# --solutions--
-												feat(challenge-md): Add initial markdown challenge files

											
										
										
											2018-09-30 22:01:58 +00:00
 								```js
 								// solution required
 								```