## Hopping Game

### Problem 391

Published on Saturday, 30th June 2012, 11:00 pm; Solved by 190; Difficulty rating: 75%
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, ...}.

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.