Cutting A Very Large Polygon
Dan and Sam play a game on a regular polygon of 100 sides. Each one draws a diagonal on the polygon in his turn.
When someone draws a diagonal, it cannot have common points (except the vertexes of the polygon) with other diagonals already drawn.
The game finishes when someone can't draw a diagonal on the polygon following the rules; that person is the loser. If Dan begins, who will win? In other words, who has a winning strategy?
Clarification: The diagonals of a polygon are straight lines that join non-adjacent vertexes.
This is the thirteenth problem of the set Winning Strategies .
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1 solution
Relevant wiki: Combinatorial Games - Winning Positions
Since 100 is even, Dan can draw a diagonal splitting the polygon in half. On subsequent turns, he will mirror Sam's move in the half that Sam did not play in. Thus, Dan can always move if Sam could, so Dan wins.