Concave -- A Game Of Triangles
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From: Leroy Quet on 5 Jul 2010 15:32 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. 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. The player gets an addition point every move he completes a triangle with only one line-segment. (Note: A player only makes one triangle per move.) The game ends when the convex hull of the dots is completed. The player with the most points wins. Thanks, Leroy Quet
From: Leroy Quet on 5 Jul 2010 15:40 Leroy Quet wrote: >... > 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. >.... Here, the dots may go anywhere on the paper the players choose except upon dots already drawn. >.... > 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. >.... The number of points in this move is the number of concave angles in the perimeter of the bounding polygon AT THE CONCLUSION of a player's move (plus the extra point, if applicable). The same angles can count again towards a players score in different moves if the angles remain along the perimeter of the bounding polygon. Thanks, Leroy Quet
From: Torben �gidius Mogensen on 6 Jul 2010 08:02 Leroy Quet <qqquet(a)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
From: Leroy Quet on 6 Jul 2010 08:25 Torben Ægidius Mogensen wrote: > Leroy Quet <qqquet(a)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"? Three dots cannot be connected such that there are any dots in the interior of the triangle. > > 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? Yes, I mean an angle greater than 180 degrees. The "concave angles", or what are actually called 'reflex angles', I am referring to are along the permeter of the bounding polygon. So, if there is at least one concave/reflex angle, then the bounding polygon is a concave polygon. > > 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. I think by "acute angles" you mean what I refer to as "concave angles", or reflex angles. Yes, drawing just one line to make a triangle deprives the player of a point from a reflex angles at the same time it gains him a point for drawing one line. But the player has also deprived the next player -- and the next, and maybe himself again on his next move -- an additional point from the reduction in reflex angles. Strategy. Thanks, Leroy Quet
From: James Waldby on 6 Jul 2010 11:48
On Tue, 06 Jul 2010 05:25:54 -0700, Leroy Quet wrote: > Torben Ægidius Mogensen wrote: >> Leroy Quet writes: .... >> > 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"? > > Three dots cannot be connected such that there are any dots in the > interior of the triangle. That phrasing is confusing. Some other problems: - Just say "in general linear position" vs no 3 dots co-linear, and refer to <http://en.wikipedia.org/wiki/General_position>. - Re the Dots part of the game, you say "In the first part of the game", leading to an expectation of a second or later part, but never explicitly denote a second-part description. - In the Lines part of the game, the first player could connect three dots with three lines such that each dot is in the interior of a line, or such that two dots are in the interior of one line and the third dot in the interior of another line. The rules as given don't prohibit doing so. But if the first player does so, play is over, with the first player getting a score of three or zero (see next item) and the other player zero. - The "reflex angle" term that you use in your reply to Torben is decidedly less ambiguous than "concave angle" but it isn't clear to me whether you are measuring angles (on the perimeter of the bounding polygon) from the exterior of the polygon or from the interior. When the first player completes the first triangle, what score does she or he or it get? See below for a rephrasing of first few paragraphs. >> > 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? > > > Yes, I mean an angle greater than 180 degrees. The "concave angles", or > what are actually called 'reflex angles', I am referring to are along > the perimeter of the bounding polygon. So, if there is at least one > concave/reflex angle, then the bounding polygon is a concave polygon. > > >> > 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. > > I think by "acute angles" you mean what I refer to as "concave angles", > or reflex angles. Yes, drawing just one line to make a triangle > deprives the player of a point from a reflex angles at the same time it > gains him a point for drawing one line. [...] A rephrasing of first few paragraphs of game description: To start, draw three dots and three straight line segments between those dots, to form an initial triangle IT on the paper. Thereafter, the game is played in two parts, Dots and Lines. In each part of the game, players take turns in sequence. For Dots, choose some number of turns per player, n. (A total of 3+n*p dots will be drawn.) On each Dots turn, a player draws one dot, outside IT. All dots, those of IT included, shall be in general linear position. After n turns each, go on to Lines. In the Lines part of the game, on each turn, a player draws one or two line segments, to form a triangle with no dots inside it. Every line segment drawn must begin at the end of a previously-drawn line segment and end at a dot. (Then describe scoring, which I don't understand, as noted before) -- jiw |