MathDB

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.
Czech Republic

C7

Problems(1)