Consider the following game.

The game state is a vector M in R^n, and is initialized as the 0 vector.

A valid move is a vector v in R^n such |v| = 1 and v >= 0

The game state after move v is s(M+v), where s(x) subtracts 1 from the largest component in x.

The score of a game state is the largest component in M.

What is the finite sequence of moves, V = v1, v2, .... vk which will maximize the score. What is the maximum possible score?

So far, I have found an upper bound of n and a lower bound of O(log n)

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