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10-gon game


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6 replies to this topic

#1 BMAD

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Posted 14 July 2013 - 02:34 AM

There is a coin at each vertex of a regular 10-gon. Alice and Bob take turns removing one coin, with Alice going first. A coin can be removed only if there is an acute-angled triangle between the coin one wants to remove and two other coins [the center of the shape is the center of the angle]. A player who cannot move loses. (Note: A 90 angle is not acute.)
 
 
Who has a winning strategy?

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#2 gavinksong

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Posted 14 July 2013 - 02:15 PM

Spoiler for


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#3 gavinksong

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Posted 14 July 2013 - 02:24 PM

Spoiler for


Edited by gavinksong, 14 July 2013 - 02:31 PM.

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#4 BMAD

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Posted 14 July 2013 - 03:18 PM

Spoiler for

remember right angled is not considered acute in the game's directions


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#5 gavinksong

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Posted 14 July 2013 - 03:50 PM

 

Spoiler for

remember right angled is not considered acute in the game's directions

 

Spoiler for


Edited by gavinksong, 14 July 2013 - 03:56 PM.

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#6 DeGe

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Posted 15 July 2013 - 01:00 PM


remember right angled is not considered acute in the game's directions

I don't see why 90° should be an issue. There are no coins that make 90° -- 10 sides for 360°; angle between any 2 vertices is a multiple of 36° and will never be 90°.

I agree with gavin on the solution.
 


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#7 gavinksong

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Posted 15 July 2013 - 01:13 PM

 


remember right angled is not considered acute in the game's directions

I don't see why 90° should be an issue. There are no coins that make 90° -- 10 sides for 360°; angle between any 2 vertices is a multiple of 36° and will never be 90°.

I agree with gavin on the solution.
 

 

The problem is talking about acute triangles (and right triangles and obtuse triangles) formed by three vertices, not angles formed by only two vertices and the center point. I actually got this confused at first too. So any triangle whose base is the diameter of a circle and whose remaining vertex lies on the circle is a right triangle. If you imagine the 10-gon as being circumscribed by a circle, you get a right triangle whenever two of its vertices are opposite each other.


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