### Brain Teasers

# Diagonals of a Dodecagon

Jim and Tom play a game.

A dodecagon (i.e., a 12-sided polygon) is drawn on a piece of paper. They take turns drawing diagonals, i.e., a line segment to connect two non-adjacent vertices. However, they are not allowed to draw a diagonal that intersects another diagonal already drawn.

If a player can't draw another diagonal, he loses and his opponent wins. Jim draws first and Tom draws second. What's the correct strategy for Jim to win?

A dodecagon (i.e., a 12-sided polygon) is drawn on a piece of paper. They take turns drawing diagonals, i.e., a line segment to connect two non-adjacent vertices. However, they are not allowed to draw a diagonal that intersects another diagonal already drawn.

If a player can't draw another diagonal, he loses and his opponent wins. Jim draws first and Tom draws second. What's the correct strategy for Jim to win?

### Hint

If you think that the correct strategy is for Jim to first draw a "longest diagonal", which divides the dodecagon into two halves, then mimic Tom's moves, you're tricked.### Answer

If a player wins, then the dodecagon must have been divided into triangles, which requires EXACTLY NINE cuts.Therefore, Jim ALWAYS wins no matter what strategy he uses!

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