Arcs And Marks
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From: Leroy Quet on 25 Feb 2010 14:05 Here is a game for any plural number of players. Needed: blank paper, pencil and compass (the circle-drawing kind), maybe a protractor. First, draw a circle on a piece of paper, relatively large. Let m be a multiple of the number of players playing this game. m should be >= 8, at least, I suggest. Draw m pencil-marks EVENLY SPACED along the circumference of the circle. (This is why you might need a protractor, if you can't just use the compass and straightedge to accomplish this by geometric construction.) There will be m moves in the game (or in the round). Players alternately take turns drawing arcs with the compass. On a move, a player uses the appropriate mark as the center of the circular arc that passes through two other marks. The two marks (which the arc intersects the main circle at) must be equidistant from the arc's center mark, and that distant must be nonzero. The first player starts at any mark to make the center of his arc; and on each move after the first, the moving player uses for her arc's center the mark immediately clockwise from the mark used for the previous player's arc-center. Count the number of preexisting arcs (not including the "arc" of the main circle) that the moving player's arc intersects or touches. Each player has a running-total of the number of arcs his/her arcs passed through. The winner has the FEWEST total number of earlier-drawn arcs (drawn by any player) intersected or touched by their arcs. Notes: An arc may touch/intersect other single arcs more than once each, but each such incident only counts once towards the number of arcs intersected or touched. Also, multiple arcs touching at a point are counted as touching all other arcs coming together at that point, no matter which arcs are "behind" or "ahead of" which other arcs. There seems to be an advantage for who moves first, and a disadvantage for who moves last. So, maybe multiple rounds should be played, a different round where each player is the first player to move. (Draw a new circle with the same number of marks each round.) Then the total scores for the rounds are added up for the grand score for each player. Players try to minimize their grand scores, of course. Thanks, Leroy Quet
From: Leroy Quet on 25 Feb 2010 14:06 I see that I was unclear. Only one arc is drawn by a player on each move. Thanks, Leroy Quet Leroy Quet wrote: > Here is a game for any plural number of players. > > Needed: blank paper, pencil and compass (the circle-drawing kind), > maybe a protractor. > > > First, draw a circle on a piece of paper, relatively large. > > Let m be a multiple of the number of players playing this game. m > should be >= 8, at least, I suggest. > > Draw m pencil-marks EVENLY SPACED along the circumference of the > circle. (This is why you might need a protractor, if you can't just > use the compass and straightedge to accomplish this by geometric > construction.) > > There will be m moves in the game (or in the round). > > Players alternately take turns drawing arcs with the compass. On a > move, a player uses the appropriate mark as the center of the circular > arc that passes through two other marks. The two marks (which the arc > intersects the main circle at) must be equidistant from the arc's > center mark, and that distant must be nonzero. > The first player starts at any mark to make the center of his arc; and > on each move after the first, the moving player uses for her arc's > center the mark immediately clockwise from the mark used for the > previous player's arc-center. > > Count the number of preexisting arcs (not including the "arc" of the > main circle) that the moving player's arc intersects or touches. Each > player has a running-total of the number of arcs his/her arcs passed > through. > > The winner has the FEWEST total number of earlier-drawn arcs (drawn by > any player) intersected or touched by their arcs. > > Notes: An arc may touch/intersect other single arcs more than once > each, but each such incident only counts once towards the number of > arcs intersected or touched. > > Also, multiple arcs touching at a point are counted as touching all > other arcs coming together at that point, no matter which arcs are > "behind" or "ahead of" which other arcs. > > There seems to be an advantage for who moves first, and a disadvantage > for who moves last. So, maybe multiple rounds should be played, a > different round where each player is the first player to move. (Draw a > new circle with the same number of marks each round.) Then the total > scores for the rounds are added up for the grand score for each > player. Players try to minimize their grand scores, of course. > > Thanks, > Leroy Quet
From: Leroy Quet on 25 Feb 2010 14:13 Another clarification: Each player's arcs must be drawn within the interior of the circle from the circle's edge back to the circle's edge. Thanks, Leroy Quet Leroy Quet wrote: > I see that I was unclear. > > Only one arc is drawn by a player on each move. > > Thanks, > Leroy Quet > > > Leroy Quet wrote: > > Here is a game for any plural number of players. > > > > Needed: blank paper, pencil and compass (the circle-drawing kind), > > maybe a protractor. > > > > > > First, draw a circle on a piece of paper, relatively large. > > > > Let m be a multiple of the number of players playing this game. m > > should be >= 8, at least, I suggest. > > > > Draw m pencil-marks EVENLY SPACED along the circumference of the > > circle. (This is why you might need a protractor, if you can't just > > use the compass and straightedge to accomplish this by geometric > > construction.) > > > > There will be m moves in the game (or in the round). > > > > Players alternately take turns drawing arcs with the compass. On a > > move, a player uses the appropriate mark as the center of the circular > > arc that passes through two other marks. The two marks (which the arc > > intersects the main circle at) must be equidistant from the arc's > > center mark, and that distant must be nonzero. > > The first player starts at any mark to make the center of his arc; and > > on each move after the first, the moving player uses for her arc's > > center the mark immediately clockwise from the mark used for the > > previous player's arc-center. > > > > Count the number of preexisting arcs (not including the "arc" of the > > main circle) that the moving player's arc intersects or touches. Each > > player has a running-total of the number of arcs his/her arcs passed > > through. > > > > The winner has the FEWEST total number of earlier-drawn arcs (drawn by > > any player) intersected or touched by their arcs. > > > > Notes: An arc may touch/intersect other single arcs more than once > > each, but each such incident only counts once towards the number of > > arcs intersected or touched. > > > > Also, multiple arcs touching at a point are counted as touching all > > other arcs coming together at that point, no matter which arcs are > > "behind" or "ahead of" which other arcs. > > > > There seems to be an advantage for who moves first, and a disadvantage > > for who moves last. So, maybe multiple rounds should be played, a > > different round where each player is the first player to move. (Draw a > > new circle with the same number of marks each round.) Then the total > > scores for the rounds are added up for the grand score for each > > player. Players try to minimize their grand scores, of course. > > > > Thanks, > > Leroy Quet
From: Mark Steere on 26 Feb 2010 08:27 On Feb 25, 11:05 am, Leroy Quet <qqq...(a)mindspring.com> wrote: > Players alternately take turns drawing arcs with the compass. Players take turns. The "alternately" is redundant.
From: Mensanator on 26 Feb 2010 14:25
On Feb 26, 7:27 am, Mark Steere <markste...(a)gmail.com> wrote: > On Feb 25, 11:05 am, Leroy Quet <qqq...(a)mindspring.com> wrote: > > > Players alternately take turns drawing arcs with the compass. > > Players take turns. The "alternately" is redundant. Not really. The rules could allow a player take take multiple, successive turns. The "alternately" speciffically prohibits this. |