The 4 bottles
There are 4 bottles, each bottle in order from left to right has 1, 2, 3, 4 marbles. A and B play a game, A puts 2 marbles, one per bottle into 2 bottles which are next to each other (A decides what are those 2 bottles), then B switches the places of 2 bottles which are next to each other (B decides what are those 2 bottles). The cycle keeps continues until A wins the game when all four bottles have the same numbers of marbles.
If A starts first, can B let A never win the game?
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1 solution
Suppose B simply does nothing. Then whenever A puts a marble in the first bottle, the number of marbles in the second one increases by one. Hence, these two bottles can never contain the same number of marbles.