100001-gon
Dan and Sam play a game on a convex polygon of 100001 sides. Each one draws a diagonal on the polygon in his turn.
When someone draws a diagonal, it cannot have common points (except the vertices 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? This means, who has a winning strategy?
Clarification: The diagonals of a polygon are straight lines that join non-adjacent vertices.
This is the fourteenth problem of the set Winning Strategies .
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When Dan draws a diagonal in the first turn, he divides the polygon in two convex polygons: one with an even number of sides and one with an odd number. Sam must split the even polygonal in half.
If in subsequent turns Dan plays in one half, Sam just mirror Dan's move in the other half, so Sam wins in the even polygon.
If Dan decides to play on the odd polygon, we're beginning the game again: he divides the polygon in two convex polygons, one with an even number of sides and one with an odd number. Anew, Sam wins in the even polygon.
As the polygon consists of a finite number of sides, after some turns, Dan will have splitted an odd polygon in a triangle and an even polygon (at least, a quadrilateral). Sam wins in the even polygon, and in a triangle nobody can draw a diagonal. Therefore, Sam wins.