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Topic: Concave -- A Game Of Triangles
Replies: 5   Last Post: Jul 7, 2010 3:23 PM

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Torben Mogensen

Posts: 60
Registered: 12/6/04
Re: Concave -- A Game Of Triangles
Posted: Jul 6, 2010 8:02 AM
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Leroy Quet <qqquet@mindspring.com> writes:

> Here is a game for any plural number of players.
>
> Needed: Blank piece of paper and a pen/pencil.
>
> In the first part of the game, players take turns placing n dots on
> the paper. No 3 dots are to be co-linear. n is determined ahead of
> time amongst the players and is a multiple of the number of players.
>
> The first player to move connects any (almost any) 3 dots with 3
> straight line segments to make a triangle, such that no dots are on
> the interior of the triangle.


Why only "almost any"?

> Thereafter, the players continue to take turns. On a turn, a player
> makes a triangle either by connecting (with two straight line
> segments) a single dot to two dots that are at the ends of the side of
> a triangle already drawn, or by drawing a single line connecting two
> dots that are two vertices of two adjacent triangles that come
> together to form a concave angle (a concave angle along the perimeter
> of the polygon containing all the triangles drawn so far).
>
> No line-segments shall coincide with line-segments already drawn.
>
> The triangles drawn in this game shall contain no dots in their
> interiors.
>
> After a player moves, she gets a point for each concave angle along
> the perimeter of the bounding polygon, the polygon containing all the
> triangles drawn so far.


My English may be a bit weak, but I'm not completely sure what you mean
by "concave angle". I know obtuse and acute angles and concave and
convex polygons, but I don't recall seing the word "concave" being used
about angles. Are they angles of more than 180 degrees?

> The player gets an addition point every move he completes a triangle
> with only one line-segment.


Would this not be the same as saying that a player scores the number of
acute angles either before or after his move, whichever gives the most
points? If a player closes a triangle with one line segment, the number
of acute angles (as I understand the term) is reduced by one, so 1 plus
the number of remaining acute angles is the same as the number of acute
angles before the move.

Torben



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