Problem 391

Hopping Game

Let `s _{k}` be the number of 1’s when writing the numbers from 0 to

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

The sequence S = {

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.

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