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

--description--

Let s_k 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 s_5 = 7.

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

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, with n = 5 and starting with c = 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.

It can be verified \sum M{(n)}^3 = 8150 for 1 ≤ n ≤ 20.

Find \sum M{(n)}^3 for 1 ≤ n ≤ 1000.

--hints--

hoppingGame() should return 61029882288.

assert.strictEqual(hoppingGame(), 61029882288);

--seed--

--seed-contents--

function hoppingGame() {

  return true;
}

hoppingGame();

--solutions--

// solution required

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