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Omni-MATH / There are 60 empty boxes B₁,…,B₆₀ in a row on a table and an unlimited supply of pebbles. Given a positive integer n, Alice and Bob play the following game. In the first round, Ali…

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problem

There are 60 empty boxes B1,…,B60B_1,\ldots,B_{60} in a row on a table and an unlimited supply of pebbles. Given a positive integer nn, Alice and Bob play the following game. In the first round, Alice takes nn pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer kk with 1≤k≤591\leq k\leq 59 and splits the boxes into the two groups B1,…,BkB_1,\ldots,B_k and Bk+1,…,B60B_{k+1},\ldots,B_{60}. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest nn such that Alice can prevent Bob from winning. [i]Czech Republic[/i]
Plain-text mathematical notation (without MathML)
There are 60 empty boxes B₁,…,B₆₀ in a row on a table and an unlimited supply of pebbles. Given a positive integer n, Alice and Bob play the following game.
In the first round, Alice takes n pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer k with 1≤k≤59 and splits the boxes into the two groups B₁,…,B_(k) and B_(k+1),…,B₆₀.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest n such that Alice can prevent Bob from winning.

[i]Czech Republic[/i]
Original LaTeX notation
There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.

[i]Czech Republic[/i]

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task_id
81869be6-3d72-5942-9267-5f56f3e84e51
task_key
test--81869be6-3d72-5942-9267-5f56f3e84e51
task_revision_id
1
config / split
/ test

Official source

initial import