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Consider a regular polygon

A_1A_2\ldots A_{6n+3}

. The vertices

A_{2n+1}, A_{4n+2}, A_{6n+3}

are called holes. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players

A

and

B

take moves in turn. In each move, starting from

A

, the player chooses pebble and puts it to the next vertex clockwise (for example,

A_2\rightarrow A_3

A_{6n+3}\rightarrow A_1

). Player

A

wins if at least two pebbles lie in holes after someone's move. Does player

A

always have winning strategy?
Proposed by Bohdan Rublov

Problem Statement